Distribution of Algebraic Numbers and Metric Theory of Diophantine Approximation

Abstract

In this paper we give an overview of recent results regarding close conjugate algebraic numbers, the number of integral polynomials with small discriminant and pairs of polynomials with small resultants.

2010 Mathematics Subject Classification. 11J83, 11J13, 11K60, 11K55

1 Introduction

Throughout the paper μA stands for the Lebesgue measure of a measurable set $A \subset \mathbb{R}$ and dim B denotes the Hausdorff dimension of B. Given $\psi : \mathbb{N} \rightarrow (0 + \infty )$, let $\mathcal{L}(\psi )$ denote the set of $x \in \mathbb{R}$ such that

$$\left \vert x -\frac{p} {q}\right \vert < \frac{\psi (q)} {q}$$

(1)

has infinitely many solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$. We begin by recalling two classical results in metric theory of Diophantine approximation.

Khintchine’s theorem [35] . Let $\psi : \mathbb{N} \rightarrow (0,+\infty )$ be monotonic and I be an interval in $\mathbb{R}$. Then

$$\mu (I\cap \mathcal{L}(\psi )) = \left \{\begin{array}{@{}l@{\quad}l@{}} \ 0,\ \ \ \quad &{if}\ \displaystyle{\sum} _{q=1}^{\infty}\psi (q) < \infty , \\ \mu (I),\ \quad &{if}\ \displaystyle{\sum}_{q=1}^{\infty }\psi (q) = \infty. \end{array} \right.$$

(2)

Jarník–Besicovitch theorem [25, 34] . Let v > 1 and for $q \in \mathbb{N}$ let $\psi _{v}(q) = {q}^{-v}$ . Then

$$\dim \,\mathcal{L}(\psi _{v}) = \frac{2} {v + 1}.$$

The condition that ψ is monotonic can be omitted from the convergence case of Khintchine’s theorem, though it is vital in the case of divergence—see [12, 33, 42] for a further discussion. By the turn of the millennium the above theorems were generalised in various directions. One important direction of research has been Diophantine approximation by algebraic numbers and/or integral polynomials, which has eventually grown into an area of number theory known as Diophantine approximation on manifolds.

Given a polynomial $P = a_{n}{x}^{n} +\cdots +a_{1}x + a_{0} \in \mathbb{Z}[x]$, the number $H = H(P) =\max _{0\leq i\leq n}\vert a_{j}\vert$ will be called the (naive) height of P. Given $n \in \mathbb{N}$ and an approximation function $\Psi : \mathbb{N} \rightarrow (0,+\infty )$, let $\mathcal{L}_{n}(\Psi )$ be the set of $x \in \mathbb{R}$ such that

$$\vert P(x)\vert < \Psi (H(P))$$

(3)

for infinitely many $P \in \mathbb{Z}[x] \setminus \{ 0\}$ with deg P ≤ n. Note that $\mathcal{L}_{1}(\Psi )$ is essentially the same as the set $\mathcal{L}(\Psi )$ introduced above. Thus, the following statement represents an analogue of Khintchine’s theorem for the case of polynomials.

Theorem 1.

Let $n \in \mathbb{N}$ and $\Psi : \mathbb{N} \rightarrow (0,+\infty )$ be monotonic. Then for any interval I

$$\mu (I\cap \mathcal{L}_{n}(\Psi )) = \left \{\begin{array}{@{}l@{\quad }l@{}} \ 0,\ \ \ \quad &{if}\ \displaystyle{\sum} _{h=1}^{\infty }{h}^{n-1}\Psi (h) < \infty , \\ \mu (I),\ \quad &{if}\ \displaystyle{\sum} _{h=1}^{\infty}{h}^{n-1}\Psi (h) = \infty. \end{array} \right.$$

(4)

The case of convergence of Theorem 1 was proved in [17], the case of divergence was proved in [4]. The condition that $\Psi$ is monotonic can be omitted from the case of convergence as shown in [6]. Theorem 1 was generalised to the case of approximation in the fields of complex and p-adic numbers [9, 19], to simultaneous approximations in $\mathbb{R} \times \mathbb{C} \times \mathbb{Q}_{p}$ [22, 26] and to various other settings. When $\Psi = \Psi _{w}$ is given by $\Psi _{w}(q) = {q}^{-w}$ Theorem 1 reduces to a famous problem of Mahler [37, 41] solved by Sprindžuk. The versions of Theorem 1 for monic polynomials were established in [27, 40]. For the more general case of Diophantine approximation on manifolds see, for example, [5, 7, 10, 15, 18, 20, 36, 42].

The more delicate Jarník-Besicovitch theorem was also generalised to the case of polynomials and reads as follows.

Theorem 2.

Let w > n and $\Psi _{w}(q) = {q}^{-w}$ . Then

$$\dim \,\mathcal{L}_{n}(\Psi _{w}) = \frac{n + 1} {w + 1}.$$

(5)

The lower bound $\dim \,\mathcal{L}_{n}(\Psi _{w}) \geq \frac{n+1} {w+1}$ was obtained by Baker and Schmidt [2] who also conjectured (5). The conjecture was proved in full generality in [16]. It is worth noting that the generalised Baker-Schmidt problem for manifolds remains an open challenging problem in dimensions n ≥ 3; the case of n = 2 was settled by R.C. Baker [3], see also [1, 8] and [7, 11, 43] for its analogue for simultaneous rational approximations.

The various techniques used to prove Theorems 1 and 2 make a substantial use of the properties of discriminants and resultants of polynomials and to some extent the distribution of algebraic numbers. The main substance of this paper will be to overview some relevant recent developments and techniques in this area.

2 Distribution of Discriminants of Integral Polynomials

The discriminant of a polynomial is a vital characteristic that crops up in various problems of number theory. For example, they play an important role in Diophantine equations, Diophantine approximation and algebraic number theory [41].

Let

$$P(x) = a_{n}{x}^{n} + a_{ n-1}{x}^{n-1} +\ldots +a_{ 1}x + a_{0}$$

be a polynomial of degree n and $\alpha _{1},\alpha _{2},\cdots ,\alpha _{n}$ be its roots. By definition, the discriminant of P is given by

$$D(P) = a_{n}^{2n-2}\displaystyle\prod _{ 1\leq i<j\leq n}{(\alpha _{i} -\alpha _{j})}^{2}.$$

(6)

The following matrix formula for D(P) is well known:

$$D(P) = {(-1)}^{n(n-1)/2}\left \vert \begin{array}{ccccccc} 1 & a_{n-1} & a_{n-2} & \ldots & a_{0} & 0 & \ldots \\ 0 & a_{n} & a_{n-1} & \ldots & a_{1} & a_{0} & \ldots \\ & & & \ldots & & & \\ 0 & \ldots & 0 &a_{n}& \ldots &a_{1} & a_{0} \\ n&(n - 1)a_{n-1} & (n - 2)a_{n-2} & \ldots & 0 & 0 & \ldots \\ 0 & na_{n} &(n - 1)a_{n-1} & \ldots & a_{1} & 0 & \ldots \\ & & & \ldots & & & \\ 0 & \ldots & \ldots & 0 &na_{n}& \ldots &a_{1}\\ \end{array} \right \vert.$$

Thus, the discriminant is an integer polynomial of the coefficients of P. Consequently, whenever P has rational integer coefficients the discriminant D(P) is also an integer. Furthermore,

$$\vert D(P)\vert \geqslant 1\quad \text{for any}\ P \in \mathbb{Z}[x]\ \mathrm{with}\ \deg P \geq 1\ \mathrm{and}\ D(P)\neq 0.$$

(7)

Clearly, by (6), D(P)≠0 if and only if P has no multiple roots.

Fix $n \in \mathbb{N}$. Let Q > Q ₀(n), where Q ₀(n) is a sufficiently large number. Let $\mathcal{P}_{n}(Q)$ denote the set of non-zero polynomials $P \in \mathbb{Z}[x]$ with degP ≤ n and H(P) ≤ Q. Throughout c _j , $j = 0,1,\ldots$ will stand for positive constants depending on n only. When it is not essential for calculations we will denote these constants as c(n). Also we will use the Vinogradov symbols: A ≪ B meaning that A ≤ c(n)B. The expression $A \asymp B$ will mean B ≪ A ≪ B. Finally #S means the cardinality of a finite set S. In what follows we consider polynomials such that

$$c_{1}Q < H(P) \leq Q,\ 0 < c_{1} < 1.$$

(8)

Using the matrix representation for D(P) one readily verifies that | D(P) |  < c(n)Q ^2n − 2 for $P \in \mathcal{P}_{n}(Q)$. Thus, by (7), we have that

$$1 \leq \vert D(P)\vert < c(n){Q}^{2n-2}$$

(9)

for polynomials $P \in \mathcal{P}_{n}(Q)$ with no multiple roots. Further, it is easily verified that

$$\#\mathcal{P}_{n}(Q) < {2}^{2n+2}{Q}^{n+1}.$$

The latter together with (9) shows that [1, c(n)Q ^2n − 2] contains intervals of length c(n)Q ^n − 3 that are not hit by the values of D(P) for any $P \in \mathcal{P}_{n}(Q)$ whatsoever. For $n \geq 4$ these intervals can be arbitrarily large. Thus, the discriminants D(P) are rather sparse in the interval [1, c(n)Q ^2n − 2].

In order to understand the distribution of the values of D(P) as P varies within $\mathcal{P}_{n}(Q)$, for each given v ≥ 0 we introduce the following subclass of $\mathcal{P}_{n}(Q)$:

$$\mathcal{P}_{n}(Q,v) =\{ P \in \mathcal{P}_{n}(Q)\ :\ \vert D(P)\vert < {Q}^{2n-2-2v}\ \text{and\,(8)\,holds}\}.$$

(10)

These subclasses are of course dependant on the choice of c ₁, but for the moment let us think of c ₁ as a fixed constant.

We initially discuss some simple techniques utilizing the theory of continued fractions that enable one to obtain non-trivial lower bounds for $\#\mathcal{P}_{n}(Q,v)$ in terms of Q and v.

The first observation concerns shifts of the variable x by integers. More precisely, if $m \in \mathbb{Z}$ then $D(P(x)) = D(P(x - m))$. The height of P(x − m) changes as m varies. It is a simple matter to see that imposing (8) on P(x − m) restricts m to at most c ₂ values. Furthermore, (8) ensures that polynomials of relatively small height cannot be in $\mathcal{P}_{n}(Q,v)$.

By (6), the fact that P belongs to $\mathcal{P}_{n}(Q,v)$ with v > 0 implies that P must necessarily have at least two close roots. This gives rise to a natural path to constructing polynomials P in $\mathcal{P}_{n}(Q,v)$—we have to make sure that they have close roots. We now describe a very special procedure that enables one to do exactly this.

Take n best approximations (convergents) $\frac{p_{j}} {q_{j}}$ to the number $\sqrt{2}$ with $k + 1 \leq j \leq k + n$ for some $k \in \mathbb{N}$. Define the polynomial

$$T_{\sqrt{2}}(x) =\displaystyle\prod _{ j=k+1}^{k+n}(q_{ j}x - p_{j})$$

of degree n. Clearly the above mentioned best approximations to $\sqrt{ 2}$ are the roots of T. Also note that the height of T is ≪ a _n , where

$$a_{n} =\displaystyle\prod _{k+1\leq i\leq k+n}q_{i}.$$

From the theory of continued fractions we know that q _j  ≤ 3q _j − 1. Thus $a_{n} \leq c(n)q_{k+1}^{n}$. On the other hand, we also know that for i < j

$$\left \vert \frac{p_{i}} {q_{i}} -\frac{p_{j}} {q_{j}}\right \vert \leq \left \vert \frac{p_{i}} {q_{i}} -\frac{p_{i+1}} {q_{i+1}}\right \vert = \frac{1} {q_{i}q_{i+1}} < \frac{1} {q_{i}^{2}}.$$

Therefore, we can estimate the following product

$$\Pi _{1} =\displaystyle\prod _{k+1\leq i<j\leq k+n}{\left \vert \frac{p_{i}} {q_{i}} -\frac{p_{j}} {q_{j}}\right \vert }^{2} \leq \displaystyle\prod _{ k+1\leq i\leq k+n-1}{\left ( \frac{1} {q_{i}^{2}}\right )}^{2(k+n-i)} \ll q_{ k+1}^{-\sigma },$$

where

$$\sigma = 2\displaystyle\sum _{i=k+1}^{k+n-1}2(k + n - i) = 4\displaystyle\sum _{\ell =1}^{n-1}\ell = 2n(n - 1).$$

and see that

$$\vert D(T_{\sqrt{2}}(x))\vert \leq a_{n}^{2n(n-1)}\Pi _{ 1} \ll q_{k+1}^{2n(n-1)}q_{ k+1}^{-\sigma } = 1.$$

This way we construct a polynomial of degree n with arbitrarily large height and discriminant as small as c(n). However, to get quantitative bounds for $\#\mathcal{P}_{n}(Q,v)$ more needs to be done. The following lemmas underpin the construction.

Lemma 1.

Let I be an interval, $I \subset \mathbb{R}$ , c ₃ and c ₄ be positive constants such that $\max \{c_{3},c_{4}\} \leq 1$ . Given a sufficiently large Q, let $\mathcal{L}_{1,Q}(c_{3},c_{4})$ be the set of x ∈ I such that the system of inequalities

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \vert qx - p\vert < c_{3}{Q}^{-1},\quad \\ 1 \leq q \leq c_{4}Q, \quad \end{array} \right.$$

(11)

has a solution in coprime $(p,q) \in \mathbb{Z} \times \mathbb{N}$ . Then for c ₃ c ₄ < λ, $0 <\lambda < \frac{1} {3}$, we have

$$\mu \mathcal{L}_{1,Q}(c_{3},c_{4}) < 3\lambda \vert I\vert.$$

Lemma 2.

Let M _n (Q) denote the set of x ∈ I such that the following n systems of inequalities

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \frac{{Q}^{-1}} {{3}^{i}{(n+1)}^{i}} < \vert q_{i}x - p_{i}\vert < \frac{{Q}^{-1}} {{3}^{i-1}{(n+1)}^{i-1}} \quad \\ {3}^{i-2}{(n + 1)}^{i-2}Q \leq q_{i} \leq {3}^{i-1}{(n + 1)}^{i-1}Q,\ 1 \leq i \leq n\quad \end{array} \right.$$

have solutions in $\{(p_{i},q_{i})\}_{i=1}^{n}$ . Then $\mu M_{n}(Q) > \frac{\vert I\vert } {n+1}$.

Lemma 1 is proved by summing up the measures of intervals given by the first inequality of (11). Lemma 2 is a corollary of Lemma 1 (which should be applied n times) and Minkowski’s theorem for convex bodies. See [21] for details.

Now take any point x ₁ ∈ M _n (Q) and define

$$T_{1}(x) =\displaystyle\prod _{ j=1}^{n}(q_{ j}x - p_{j})\qquad \text{and}\qquad Q = c(n)\displaystyle\prod _{i=1}^{n}q_{ i},$$

where (p _j , q _j ) arise from Lemma 2. Estimating | D(T ₁) | gives

$$\vert D(T_{1})\vert \ll {Q}^{2n-2}\displaystyle\prod _{ 1\leq i<j\leq n}\left \vert \frac{p_{i}} {q_{i}} -\frac{p_{j}} {q_{j}}\right \vert \ll 1.$$

We now use the fact that M _n (Q) is a fairly large subset of I to produce other polynomials with this property. For this purpose we choose points $x_{2},x_{3},\ldots \in M_{n}(Q)$ that are well separated. As a result we obtain

Theorem 3 ([21]). 

For any sufficiently large Q there are $c(n){Q}^{\frac{2} {n} }$ polynomials $P \in \mathcal{P}_{n}(Q)$ such that 1 ≤|D(P)|≤ c(n).

The above ideas can be generalised to give a similar bound for the number of polynomials $P \in \mathcal{P}_{n}(Q)$ such that | D(P) | lies in a neighborhood of some K with c(n) < K < c(n)Q ^2n − 2.

Theorem 4 ([21]). 

For any θ, 0 ≤θ ≤ 2n − 2, there are at least c(n)Q ^2∕n polynomials $P \in \mathcal{P}_{n}(Q)$ with discriminants satisfying the inequalities

$$c_{5}{Q}^{\theta } < \vert D(P)\vert < c_{6}{Q}^{\theta }.$$

We proceed by describing a more sophisticated method from [23] that produces lower bounds for $\#\mathcal{P}_{n}(Q,v)$. The main result is as follows.

Theorem 5 ([23]). 

Let $v \in [0, \frac{1} {2}]$ . Then there are at least $c(n){Q}^{n+1-2v}$ polynomials $P \in \mathcal{P}_{n}(Q)$ with discriminants

$$\vert D(P)\vert < {Q}^{2n-2-2v}.$$

(12)

Establishing upper bounds for $\#\mathcal{P}_{n}(Q,v)$ is likely a more difficult task. We expect that if we impose some reasonable conditions on polynomials P from $\mathcal{P}_{n}(Q)$ (for example excluding reducible polynomials) then the lower bound given by Theorem 5 would become sharp. We now state this formally as the following

Problem 1.

Find reasonable constrains on polynomials P that chop a subclass $\mathcal{P}{^\prime}_{n}(Q)$ off $\mathcal{P}_{n}(Q)$ such that $\#\mathcal{P}{^\prime}(Q) \asymp \#\mathcal{P}_{n}(Q)$ and for $v \in [0, \tfrac{1} {2}]$

$$\#\{P \in \mathcal{P}{^\prime}_{n}(Q) : \vert D(P)\vert < {Q}^{2(n-1-v)}\} \asymp {Q}^{n+1-2v}.$$

(13)

Obtaining the estimates of this ilk for a larger range of v is another problem. We wish to note that (13) is false for $\mathcal{P}_{n}(Q)$—see [32] for precise upper and lower bounds in the case v < 3 ∕ 5 and n = 3.

Problem 2.

For each n find the function f _n (v), if it exists at all, such that for all sufficiently large Q one has the estimates

$$\#\{P \in \mathcal{P}_{n}(Q) : \vert D(P)\vert < {Q}^{2(n-1-v)}\} \asymp {Q}^{n+1-f_{n}(v)}.$$

(14)

It was shown in [32] that $f_{3}(v) = \tfrac{5} {3}v$ for 0 ≤ v ≤ 3 ∕ 5.

2.1 Sketch of the Proof of Theorem 5

Underlying the proof of Theorem 5 is the following result, which essentially plays the role of Lemma 1 in this more general context. In what follows, given an interval $I \subset [-1/2,1/2]$, let $\mathcal{L}_{n}(I,Q,v,c_{7},c_{8})$ be the set of x ∈ I such that

$$\left \{\begin{array}{l} \vert P(x)\vert < c_{7}{Q}^{-n+v}, \\ \vert P{^\prime}(x)\vert < c_{8}{Q}^{1-v} \end{array} \right.$$

(15)

holds for some $P \in \mathcal{P}_{n}(Q)$.

Theorem 6.

Let Q denote a sufficiently large number, $v \in [0, \frac{1} {2}]$ and let c ₇ and c ₈ be positive constants such that $c_{7}c_{8} < {n}^{-1}{2}^{-n-12}$ . Then

$$\mu \mathcal{L}_{n}(I,Q,v,c_{7},c_{8}) < \frac{\vert I\vert } {4}.$$

We now explain the role of Theorem 6 in establishing Theorem 5. Suppose that $P \in \mathbb{Z}[x]$, degP ≤ n, | a _n  |  > cH. If | a _n  | ≤ cH then the polynomial can be transformed into one with a large leading coefficient with the same discriminant—see [41].

By Dirichlet’s pigeonhole principle, for any point x ∈ I and Q > 1 the following system

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \vert P(x)\vert < {Q}^{-n+v},\quad \\ \vert P{^\prime}(x)\vert < 8n{Q}^{1-v}\quad \end{array} \right.$$

(16)

holds for some polynomial $P \in \mathcal{P}_{n}(Q)$. Let $\gamma = {n}^{-2}{2}^{-n-15}$ and $I = [-1/2,1/2]$. Then, by Theorem 6, the set

$$B_{1} = I \setminus \mathcal{L}_{n}(I,Q,v,1,\gamma ) \cup \mathcal{L}_{n}(I,Q,v,\gamma ,8n)$$

satisfies $\mu B_{1} \geqslant \frac{1} {2}$ for all sufficiently large Q. Hence for any x ₁ ∈ B ₁ the solution $P \in \mathcal{P}_{n}(Q)$ to the system (16) must satisfy

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \gamma {Q}^{-n+v} < \vert P(x_{ 1})\vert < {Q}^{-n+v},\quad \\ \gamma {Q}^{1-v} < \vert P{^\prime}(x_{1})\vert < 8n{Q}^{1-v}. \quad \end{array} \right.$$

(17)

For all x in the interval $\vert x - x_{1}\vert < {Q}^{-\frac{2} {3} }$, the Mean Value Theorem gives

$$P{^\prime}(x) = P{^\prime}(x_{1}) + P{^\prime}{^\prime}(\xi _{1})(x - x_{1})\ \ \text{for some}\ \xi _{1} \in [x,x_{1}].$$

(18)

The obvious estimate $\vert P{^\prime}{^\prime}(\xi _{2})\vert < {n}^{3}Q$ implies $\vert P{^\prime}{^\prime}(\xi _{1})(x - x_{1})\vert < {n}^{3}{Q}^{\frac{1} {3} }$. But $\vert P{^\prime}(x_{1})\vert \gg {Q}^{\frac{1} {2} }$ for $v \leq \frac{1} {2}$ and therefore, by (18) and the second inequality of (17), for sufficiently large Q we have that

$$\tfrac{\gamma } {2}{Q}^{1-v} < \tfrac{1} {2}\vert P{^\prime}(x_{1})\vert < \vert P{^\prime}(x)\vert < 2\vert P{^\prime}(x_{1})\vert < 16n{Q}^{1-v}.$$

There are four possible combinations for signs of P(x ₁) and P′(x ₁). To illustrate the ideas we consider the case when P ₁(x ₁) < 0 and P′ ₁(x ₁) > 0—the others are dealt with in a similar way. Our goal for now is to find a root of P close to x ₁. Once again we appeal to the Mean Value Theorem:

$$P(x) = P(x_{1}) + P{^\prime}(\xi _{2})(x - x_{1})\ \ \text{for some}\ \xi _{2} \in [x_{1},x].$$

(19)

Write $x = x_{1} + \Delta$ and suppose that $\Delta > {2\gamma }^{-1}{Q}^{-n-1+2v}$. If $P(x_{1}) < P(x_{1} + \Delta ) < 0$ then the first inequality of (17) implies

$$0 < P(x_{1} + \Delta ) - P(x_{1}) < {Q}^{-n+v}.$$

On the other hand we have

$$\vert P{^\prime}(\xi _{2})\Delta \vert > \frac{\gamma } {2}{Q}^{1-v}{2\gamma }^{-1}{Q}^{-n-1+2v} = {Q}^{-n+v}.$$

Thus in view of (19) we obtain a contradiction. This means that $P_{1}(x_{1} + \Delta ) > 0$ and there is a real root α of the polynomial P(x) between x ₁ and $x_{1} + \Delta$. Once again using the Mean Value Theorem and the estimates for P(x) and P′(α) we get

$$\vert x_{1} -\alpha \vert < {2\gamma }^{-1}{Q}^{-n-1+2v} = n{2}^{n+13}{Q}^{-n-1+2v}.$$

(20)

Note that as well as ensuring that α, a root of P, is close to x ₁ inequalities (17) keep α sufficiently away from x ₁. We now explain this more formally. Again we consider only one of the four possibilities: P(x ₁) > 0, P′(x ₁) < 0. With $x = x_{1} + \Delta _{1}$, by the Mean Value Theorem, we have

$$P(x) = P(x_{1}) + P{^\prime}(\xi _{3})\Delta _{1},\ \xi _{3} \in [x_{1},x].$$

(21)

If $\Delta _{1} < {2}^{-4}{n}^{-1}\gamma {Q}^{-n-1+2v}$ then in (21) the following holds: $\vert P(x_{1})\vert >\gamma {Q}^{-n+v}$ and $\vert P{^\prime}(\xi _{3})\Delta _{1}\vert <\gamma {Q}^{-n+v}$. It implies that the polynomial P(x) cannot have any root in the interval $[x_{1},x_{1} + \Delta _{1}]$ and therefore for any root α, we have

$${n}^{-1}{2}^{-n-13}{Q}^{-n-1+2v} < \vert x -\alpha \vert.$$

This time let α be the root of P closest to x ₁. By the Mean Value Theorem,

$$P{^\prime}(\alpha ) = P{^\prime}(x_{1}) + P{^\prime}{^\prime}(\xi _{4})(x_{1}-\alpha ),\ \xi _{4} \in [x,\alpha ],$$

the estimate | P′(ξ) |  < n ³ Q and (20) for sufficiently large Q we get

$${n}^{-1}{2}^{-n-13}{Q}^{1-v} < \vert P{^\prime}(\alpha )\vert < 16n{Q}^{1-v}.$$

The square of derivative is a factor of the discriminant of P. Taking into account that for $\vert a_{n}\vert \asymp H(P)$ all roots of the polynomial are bounded, see [41]. Then we can estimate the differences | α _i  − α _j  | , 2 ≤ i < j ≤ n, by a constant c(n). This way we obtain (12). Since μB ₁ ≥ 1 ∕ 2 and x ₁ is an arbitrary point in B ₁ we must have $\gg {Q}^{n+1-2v}$ different α’s that arise from (20). Since each polynomial P of degree ≤ n has at most n roots this gives $\gg {Q}^{n+1-2v}$ polynomials in $\mathcal{P}_{n}(Q,v)$ satisfying (12)—see [13] for further details.

2.2 Sketch of the Proof of Theorem 6

The purpose of this section is to discuss the key ideas of the proof of Theorem 6 given in [13] as they may be useful in a variety of other tasks. We start by estimating the measure of x such that the system

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \ \ \ \ \ \ \ \ \ \ \ \ \vert P(x)\vert < c_{11}{Q}^{-n+v}, \quad \\ {Q}^{1-v_{1}} < \vert P{^\prime}(x)\vert < c_{12}{Q}^{1-v}\quad \end{array} \right.$$

(22)

is solvable for $P \in \mathcal{P}_{n}(Q)$, where v ₁ satisfies v < v ₁ ≤ 1 and will be specified later.

We shall see that P′(x) can be replaced with P′(α) in the second inequality of (22), where α denotes the root of P nearest to x. Indeed, using the Mean Value Theorem gives

$$P{^\prime}(x) = P{^\prime}(\alpha ) + P{^\prime}{^\prime}(\xi _{1})(x-\alpha ),\ \ \ \xi _{1} \in (\alpha ,x).$$

We apply the following inequality for | x − α | 

$$\vert x -\alpha \vert < n\frac{\vert P(x)\vert } {\vert P{^\prime}(x)\vert },$$

which was proved in [17, 41]. Then

$$\vert P{^\prime}(\alpha )\vert = \vert P{^\prime}(x) - P{^\prime}{^\prime}(\xi _{5})(x-\alpha )\vert ,\ \xi _{5} \in (\alpha ,x).$$

As

$$\vert P{^\prime}{^\prime}(\xi _{1})(x-\alpha )\vert \leq {n}^{3}Qc_{ 11}n{Q}^{-n-1+v+v_{1} } = c_{11}{n}^{4}{Q}^{-n+v+v_{1} }$$

for sufficiently large Q we obtain

$$\frac{3} {4}{Q}^{1-v_{1} } \leq \frac{3} {4}\vert P{^\prime}(x)\vert \leq \vert P{^\prime}(\alpha )\vert \leq \frac{4} {3}\vert P{^\prime}(x)\vert \leq \frac{4} {3}c_{12}{Q}^{1-v}$$

and

$$\frac{3} {4}\vert P{^\prime}(\alpha )\vert \leq \vert P{^\prime}(x)\vert \leq \frac{4} {3}\vert P{^\prime}(\alpha )\vert.$$

Therefore for sufficiently large Q inequality (22) implies

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \vert P(x)\vert < c_{11}{Q}^{-n+v} \quad \\ \frac{3} {4}{Q}^{1-v_{1}} < \vert P{^\prime}(\alpha )\vert < \frac{4} {3}c_{12}{Q}^{1-v}\quad \\ \vert a_{j}\vert \leq Q. \quad \end{array} \right.$$

(23)

Let $\mathcal{L}_{n}{^\prime}(v)$ denote the set of x, for which system (23) is solvable for $P \in \mathcal{P}_{n}(Q)$. Now we are able to prove that $\mu \mathcal{L}_{n}{^\prime}(v) < \frac{3} {8}\vert I\vert$.

Consider the intervals:

$$\sigma _{1}(P) =\{ x : \vert x -\alpha \vert < \frac{4} {3}c_{11}n{Q}^{-n+v}\vert P{^\prime}(\alpha ){\vert }^{-1}\}$$

and

$$\sigma _{2}(P) =\{ x : \vert x -\alpha \vert < c_{13}{Q}^{-1+v}\vert P{^\prime}(\alpha ){\vert }^{-1}\}.$$

The value of c ₁₃ will be specified below. Of course, each polynomial P has up to n roots and potentially we have to consider all the different intervals σ ₁(P) and σ ₂(P) that correspond to each P. However, this will only affect the constant in the estimates. Thus, without loss of generality we confine ourselves to a single choice of σ ₁(P) and σ ₂(P). Obviously

$$\vert \sigma _{1}(P)\vert \leq \tfrac{4} {3}c_{11}c_{13}^{-1}n{Q}^{-n+1}\vert \sigma _{ 2}(P)\vert.$$

(24)

Fix a vector $\overline{b} = (a_{n},\ldots ,a_{2})$ of the coefficients of P. The polynomials $P \in \mathcal{P}_{n}(Q)$ with the same vector $\bar{b}$ form a subclass of $\mathcal{P}_{n}(Q)$ which will be denoted by $\mathcal{P}(\overline{b})$.

The interval σ ₂(P ₁) with $P_{1} \in \mathcal{P}(\overline{b})$ is called inessential if there is another interval σ ₂(P ₂) with $P_{2} \in \mathcal{P}(\overline{b})$ such that

$$\vert \sigma _{2}(P_{1}) \cap \sigma _{2}(P_{2})\vert \geq \tfrac{1} {2}\vert \sigma _{2}(P_{1})\vert.$$

Otherwise for any $P_{2} \in \mathcal{P}(\overline{b})$ different from P ₁

$$\vert \sigma _{2}(P_{1}) \cap \sigma _{2}(P_{2})\vert < \tfrac{1} {2}\vert \sigma _{2}(P_{1})\vert $$

and the interval σ ₂(P ₂) is called essential.

The case of essential intervals. In this case every point x ∈ I belongs to at most two essential intervals σ ₂(P). Hence for any vector $\overline{b}$

$$\displaystyle\sum _{ \begin{array}{c}P\in \mathcal{P}(\overline{b}) \\ \sigma _{2}(P)\text{ is essential}\end{array}}\vert \sigma _{2}(P)\vert \leq 2\vert I\vert.$$

(25)

The number of all possible vectors $\overline{b}$ is at most ${(2Q + 1)}^{n-1} < {2}^{n}{Q}^{n-1}.$ Then, by (24) and (25), we obtain

$$\displaystyle\sum _{\overline{b}}\displaystyle\sum _{ \begin{array}{c}P\in \mathcal{P}(\overline{b}) \\ \sigma _{2}(P)\text{ is essential}\end{array}}\vert \sigma _{1}(P)\vert < \tfrac{4} {3}c_{11}c_{13}^{-1}n{Q}^{-n+1}2\vert I\vert {2}^{n}{Q}^{n-1} = n{2}^{n+2}c_{ 11}c_{13}^{-1}\vert I\vert.$$

Thus for $c_{13} = n{2}^{n+5}c_{11}$ the measure will be not larger than $\frac{1} {8}\vert I\vert$.

The case of inessential intervals. In this case we need to estimate the values of | P _j (x) | , j = 1, 2, for $x \in \sigma _{2}(P_{1}) \cap \sigma _{2}(P_{2})$. By Taylor’s formula,

$$P_{j}(x) = P_{j}{^\prime}(\alpha )(x-\alpha ) + \tfrac{1} {2}P_{j}{^\prime}{^\prime}(\xi _{6}){(x-\alpha )}^{2}\ \text{for some}\ \xi _{ 6} \in (\alpha ,x),$$

where α is the root of either P ₁ or P ₂ as appropriate, and

$$P{^\prime}_{j}(x) = P{^\prime}_{j}(\alpha ) + P_{j}{^\prime}{^\prime}(\xi _{7})(x-\alpha )\ \text{for some}\ \xi _{7} \in (\alpha ,x).$$

The second summand is estimated by

$$\vert P{^\prime}{^\prime}(\xi _{6}){(x-\alpha )}^{2}\vert \leq 2{n}^{3}c_{ 13}^{2}{Q}^{-3+2v+2v_{1} },$$

while

$$\vert P{^\prime}(\alpha )(x-\alpha )\vert < c_{13}{Q}^{-1+v}.$$

As 2v ₁ < 2 − v for an appropriate choice of $v_{1} < \frac{3} {4}$ we obtain

$$\vert P_{j}(x)\vert \leq \tfrac{7} {6}c_{13}{Q}^{-1+v},\ j = 1,2.$$

(26)

Similarly we obtain the following estimate for P′ _j (x) when v ₁ ≤ 2 − 2v :

$$\vert P_{j}{^\prime}(x)\vert \leq \tfrac{4} {3}c_{12}{Q}^{1-v},\ j = 1,2.$$

(27)

Let $K(x) = P_{2}(x) - P_{1}(x) \in \mathbb{Z}[x]$. Obviously K(x) is non-zero and has the form $K(x) = b_{1}x + b_{0}$. By (26) and (27), we readily obtain that

$$\vert b_{1}x + b_{0}\vert < \tfrac{8} {3}c_{13}{Q}^{-1+v}$$

(28)

and

$$\vert b_{1}\vert = \vert K{^\prime}(x)\vert < \tfrac{8} {3}c_{12}{Q}^{1-v}.$$

(29)

Thus, the union of inessential intervals can be covered by intervals $\Delta (b_{1},b_{0}) \subset I$ given by (28). For fixed b ₀ and b ₁ the length of $\Delta (b_{1},b_{0})$ is bounded by $\frac{16} {3} c_{13}{Q}^{-1+v}b_{ 1}^{-1}$. Given that x ∈ I and (28) is satisfied we conclude that b ₀ takes at most | I | | b ₁ |  + 2 values. Then

$$\displaystyle\sum _{b_{0}}\vert \Delta (b_{1},b_{0})\vert \leq \tfrac{16} {3} c_{13}{Q}^{-1+v}b_{ 1}^{-1}(\vert I\vert \vert b_{ 1}\vert + 2) < 6c_{13}{Q}^{-1+v}\vert I\vert.$$

(30)

Using (29) we further obtain that

$$\displaystyle\sum _{b_{1}}\displaystyle\sum _{b_{0}}\vert \Delta (b_{1},b_{0})\vert \leq {2}^{5}c_{ 12}c_{13}{Q}^{1-v-1+v}\vert I\vert = n{2}^{n+8}c_{ 11}c_{12}\vert I\vert = \tfrac{1} {8}\vert I\vert $$

for we have that $c_{11}c_{12} < {n}^{-1}{2}^{-n-11}$. Finally, combining the estimates for essential and inessential intervals we obtain $\frac{1} {4}\vert I\vert$ as an upper bound for their total measure. The case v ≥ v ₁ can be deal with using methods described in [17] and [36].

3 Divisibility of Discriminants by Prime Powers

Let p be a prime number. Throughout μ _p denotes the Haar measure on $\mathbb{Q}_{p}$ normalized to have $\mu _{p}(\mathbb{Z}_{p}) = 1$. In this section we consider the divisibility of the discriminant D(P), $P \in \mathcal{P}_{n}(Q)$ by prime powers p ^l. This natural arithmetical question has usual interpretation in terms of Diophantine approximation in $\mathbb{Q}_{p}$, the field of p-adic number. Indeed, p ^l | D(P) if and only if | D(P) |  _p  ≤ p ^− l, where |  ⋅ |  _p stands for the p-adic norm. Thus, the question we outlined above becomes a p-adic analogue of the problems we have considered in the previous section. Naturally, we proceed with the following p-adic analogue of Theorem 5.

Theorem 7 ([24]). 

Let $v \in [0, \frac{1} {2}]$ . Then there are at least $c(n){Q}^{n+1-2v}$ polynomials $P \in \mathcal{P}_{n}(Q)$ with

$$\vert D(P)\vert _{p} < {Q}^{-2v}.$$

(31)

The proof of this result relies on the following p-adic version of Theorem 6.

Theorem 8 ([24]). 

Let Q denote a sufficiently large number and c ₁₄ and c ₁₅ denote constants depending only on n. Also, let K be a disc in $\mathbb{Q}_{p}$ . Assume that $c_{14}c_{15} < {2}^{-n-11}{p}^{-8}$ and $v \in [0, \frac{1} {2}]$ . If $M_{n,Q}(c_{14},c_{15})$ is the set of $w \in K \subset \mathbb{Q}_{p}$ such that the system of inequalities

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \vert P(w)\vert _{p} < c_{14}{Q}^{-n-1+v},\quad \\ \vert P{^\prime}(w)\vert _{p} < c_{15}{Q}^{-v} \quad \end{array} \right.$$

has solutions in polynomials $P \in \mathcal{P}_{n}(Q)$ , then

$$\mu _{p}(M_{n,Q}(c_{14},c_{15})) < \tfrac{1} {4}\mu _{p}(K).$$

The techniques used in the proof of this theorem are essentially the p-adic analogues of those used for establishing Theorem 6 and draw on the estimates obtained in [39]—see [24] for more details. Skipping any explanation of the proof of Theorem 8, we now show how it is used for establishing Theorem 7.

Let K be a disc in $\mathbb{Q}_{p}$, M _n, Q be the same as in Theorem 8, $\gamma = {p}^{-11}{2}^{-n-11}$ and

$$B = K \setminus \big (M_{n,Q}(\gamma ,{p}^{3}) \cup M_{ n,Q}(1,\gamma )\big).$$

Then, by Theorem 8, $\mu _{p}(B) \geq \frac{1} {2}\mu _{p}(K)$. Take any w ₁ ∈ B. Then, using Dirichlet’s pigeonhole principle we can find a polynomial $P \in \mathcal{P}_{n}(Q)$ such that $\vert P(w_{1})\vert _{p} < {Q}^{-n-1+v}$ and $\vert P{^\prime}(w_{1})\vert _{p} < {p}^{3}{Q}^{-v}$. Since w ₁ ∈ B we have that

$$\left \{\begin{array}{@{}l@{\quad }l@{}} \gamma {Q}^{-n-1+v} \leq \vert P(w_{1})\vert _{p} < {Q}^{-n-1+v},\quad \\ \gamma {Q}^{-v} \leq \vert P{^\prime}(w_{1})\vert _{p} < {p}^{3}{Q}^{-v}. \quad \end{array} \right.$$

(32)

Let $w \in K_{1} =\{ w \in \mathbb{Q}_{p} : \vert w - w_{1}\vert _{p} < {Q}^{-\frac{3} {4} }\}$. By Taylor’s formula

$$P{^\prime}(w) = P{^\prime}(w_{1}) +\displaystyle\sum _{ i=2}^{n}\frac{{P}^{(i)}(w_{ 1}){(w - w_{1})}^{i-1}} {(i - 1)!}.$$

Since

$$\vert (i - 1)!\vert _{p}^{-1}\vert {P}^{(i)}(w_{ 1})\vert _{p}\vert w - w_{1}\vert _{p}^{i-1} \ll {Q}^{-\frac{3} {4} },$$

and

$$\vert P{^\prime}(w)\vert _{p} \geqslant \gamma {Q}^{-v} \gg {Q}^{-\frac{1} {2} },$$

for all w ∈ K ₁ we obtain that $\vert P{^\prime}(w)\vert _{p} = \vert P{^\prime}(w_{1})\vert _{p}$. Let α be the closest root of P(w) to the point w ₁. Then, using the Mean Value Theorem, we get that $\vert w_{1} -\alpha \vert _{p} \leq \vert P(w_{1})\vert _{p}\vert P{^\prime}(w_{1})\vert _{p}^{-1}$. By (32),

$$\vert w_{1} -\alpha \vert _{p} {\leq \gamma }^{-1}{Q}^{-n-1+2v}.$$

(33)

To estimate the distance between w ₁ and the root of the polynomial we can also apply Hensel’s lemma. Since $\vert P(w_{1})\vert _{p} < \vert P{^\prime}(w_{1})\vert _{p}^{2}$ we obtain that the sequence $w_{n+1} = w_{n} -\frac{P_{1}(w_{n})} {P{^\prime}_{1}(w_{n})}$ converges to the root α ₁ of $P$ that lies in $\mathbb{Q}_{p}$ and satisfies the inequality

$$\vert w_{1} -\alpha _{1}\vert _{p} \leq \vert P(w_{1})\vert _{p}\vert P{^\prime}(w_{1})\vert _{p}^{-2} {\leq \gamma }^{-2}{Q}^{-n-1+3v}.$$

(34)

Since 0 < γ < 1 and v > 0 the right hand side of (33) is less than that of (34). This implies that the root α belongs to the disc with center w ₁ of radius less than the radius for the disc defined for the root α ₁. By Hensel’s lemma, we find that $\alpha _{1} \in \mathbb{Q}_{p}$ but estimate (33) does not guarantee that $\alpha \in \mathbb{Q}_{p}$. Suppose that α≠α ₁ and consider the discriminant of the polynomial $P \in \mathcal{P}_{n}(Q)$

$$D(P) = a_{n}^{2n-2}\displaystyle\prod _{ 1\leq i<j\leq n}{(\alpha _{i} -\alpha _{j})}^{2}.$$

(35)

From | α _j  |  _p  ≪ 1 follows that $\vert \alpha _{i} -\alpha _{j}\vert _{p} \ll 1$. The product in (35) contains the difference (α − α ₁) for some i≠j. We have $D(P) \in \mathbb{Z}$ and | D(P) | ≪ Q ^2n − 2. Assume for the moment that D(P)≠0. Then $\vert D(P)\vert _{p} \geq \vert D(P){\vert }^{-1} \gg {Q}^{-2n+2}$. From (33) and (34) we further obtain that

$$\vert \alpha _{1} -\alpha \vert _{p} = \vert \alpha _{1} - w_{1} + w_{1} -\alpha \vert _{p} \leq \max \{\vert w_{1} -\alpha _{1}\vert _{p},\vert w_{1} -\alpha \vert _{p}\} {\leq \gamma }^{-2}{Q}^{-n-1+3v}.$$

Therefore

$${Q}^{-2n+2} \ll \vert D(P)\vert _{ p} \ll \vert \alpha _{1} -\alpha \vert _{p}^{2} {<\gamma }^{-4}{Q}^{-2n-2+6v}.$$

(36)

For $v \leq \frac{1} {2}$ and Q > Q ₀ the inequality ${Q}^{-2n+2} {\ll \gamma }^{-4}{Q}^{-2n-2+6v}$ is a contradiction. Hence, α ₁ = α. Thus, $\alpha \in \mathbb{Q}_{p}$ and | w ₁ − α |  _p satisfies condition (33).

In the case when D(P) = 0 one has to use the above argument with P replaced by its factor, say $\tilde{P}$. If α and α ₁ are conjugate over $\mathbb{Q}$ one can take $\tilde{P}$ to be the minimal polynomials (over $\mathbb{Z}$) of α. Otherwise, $\tilde{P}$ is taken to be the product of the minimal polynomials for α and α ₁.

By Taylor’s formula,

$$P{^\prime}(\alpha ) = P{^\prime}(w_{1}) +\displaystyle\sum _{ i=2}^{n}{((i - 1)!)}^{-1}{P}^{(i)}(w_{ 1}){(\alpha -w_{1})}^{i-1}.$$

(37)

Observe that

$$\vert (i - 1)!\vert _{p}^{-1}\vert {P}^{(i)}(w_{ 1})\vert _{p}\vert \alpha - w_{1}\vert _{p}^{i-1} \ll {Q}^{-n-1+2v}.$$

Then, by (33), we obtain

$$\vert P{^\prime}(\alpha )\vert _{p} = \vert P{^\prime}(w_{1})\vert _{p} < {p}^{3}{Q}^{-v}.$$

Therefore

$$\vert D(P)\vert _{p} = \vert a_{n}^{2}\displaystyle\prod _{ k=2}^{n}{(\alpha _{ 1} -\alpha _{k})}^{2}\vert _{ p}\,\vert a_{n}^{2n-4}\displaystyle\prod _{ 2\leq i<j\leq n}{(\alpha _{i} -\alpha _{j})}^{2}\vert _{ p} \ll \vert P{^\prime}(\alpha )\vert _{p}^{2} \ll {Q}^{-2v}.$$

(38)

Inequality (33) implies that in the neighborhood of the point w ₁ ∈ B there exists a root α of the polynomial P with discriminant satisfying (38).

By (33), w ₁ lies in the disc $K(\alpha ,c(n){Q}^{-n-1+2v})$. Since w ₁ is an arbitrary point of B and $\mu _{p}(B) \geq \frac{1} {2}\mu _{p}(K)$, we must have $\geq c(n){Q}^{n+1-2v}\mu _{p}(K)$ discs $K(\alpha ,c(n){Q}^{-n-1+2v})$ to cover B, where α is a root of some $P \in \mathcal{P}_{n}(Q)$ satisfying (38). Since each polynomial $P \in \mathcal{P}_{n}(Q)$ has at most n roots we must have $\geq c(n){Q}^{n+1-2v}$ different polynomials $P \in \mathcal{P}_{n}(Q)$ satisfying (38), that is (31).

4 Close Conjugate Algebraic Numbers

Estimating the distance between conjugate algebraic numbers of degree n (in $\mathbb{C}$) has been investigated over the last 50 years. There are various upper and lower bounds found. However, the exact answers are known in the case of degree 2 and 3 only. Define κ _n (respectively κ _n ^∗) to be the infimum of κ such that the inequality

$$\vert \alpha _{1} -\alpha _{2}\vert > H{(\alpha _{1})}^{-\kappa }$$

holds for arbitrary conjugate algebraic numbers (respectively algebraic integers) α ₁ ≠ α ₂ of degree n with sufficiently large height H(α ₁). Here and elsewhere H(α) denotes the height of an algebraic number α, which is the absolute height of the minimal polynomial of α over $\mathbb{Z}$. Clearly, $\kappa _{n}^{{\ast}} \leq \kappa _{n}$ for all n.

In 1964 Mahler [37] proved the upper bound κ _n  ≤ n − 1, which is the best estimate up to date. It can be easily shown that κ ₂ = 1 (see, e.g. [30]). Evertse [31] proved that κ ₃ = 2. In the case of algebraic integers κ ₂ ^∗ = 0 and κ ₃ ^∗ ≥ 3 ∕ 2. The latter has been proved by Bugeaud and Mignotte [30].

For n > 3 estimates for κ _n are less satisfactory. At first Mignotte [38] showed that $\kappa _{n},\kappa _{n}^{{\ast}}\geq n/4$ for all n ≥ 3. Subsequently Bugeaud and Mignotte [29, 30] proved that

$$\begin{array}{lcl} \kappa _{n} \geq n/2 &&\text{when}\ n \geq 4\ \text{ is even,} \\ \kappa _{n}^{{\ast}} \geq (n - 1)/2&&\text{when}\ n \geq 4\ \text{ is even,} \\ \kappa _{n} \geq (n + 2)/4 &&\text{when}\ n \geq 5\ \text{ is odd,} \\ \kappa _{n}^{{\ast}} \geq (n + 2)/4&&\text{when}\ n \geq 5\ \text{ is odd.}\end{array}$$

In a recent paper Bugeaud and Dujella [28] have further shown that

$$\kappa _{n} \geq \frac{n} {2} + \frac{n - 2} {4(n - 1)}.$$

(39)

On taking an alternative route it has been shown in [14] that there are numerous examples of close conjugate algebraic numbers:

Theorem 9 ([13, 14]). 

For any n ≥ 2 we have that

$$\min \{\kappa _{n},\kappa _{n+1}^{{\ast}}\}\geq \frac{n + 1} {3}.$$

(40)

There are at least $c(n){Q}^{\frac{n+1} {3} }$ pairs of conjugate algebraic numbers of degree n (or conjugate algebraic integers of degree n + 1) α ₁ and α ₂ such that

$$\vert \alpha _{1} -\alpha _{2}\vert \asymp H{(\alpha _{1})}^{-\frac{n+1} {3} }$$

The proof of Theorem 9 is based on solvability of system of Diophantine inequalities for analytic functions [7] on the set of positive density on any interval $J \subset [-\frac{1} {2}, \frac{1} {2}]$. The interval $[-\frac{1} {2}, \frac{1} {2}]$ is taken to simplify the calculation of constants.

As above μ will denote Lebesgue measure in $\mathbb{R}$ while λ will be a non-negative constant. Given an interval $J \subset \mathbb{R}$, | J | will denote the length of J. Also, B(x, ρ) will denote the interval in $\mathbb{R}$ centered at x of radius ρ.

Let n ≥ 2 be an integer, λ ≥ 0, 0 < ν < 1 and Q > 1. Let $\mathbb{A}_{n,\nu }(Q,\lambda )$ be the set of algebraic numbers $\alpha _{1} \in \mathbb{R}$ of degree n and height H(α ₁) satisfying

$$\nu Q \leq H(\alpha _{1}) {\leq \nu }^{-1}Q$$

(41)

and

$$\nu \,{Q}^{-\lambda }\leq \vert \alpha _{ 1} -\alpha _{2}\vert {\leq \nu }^{-1}{Q}^{-\lambda }\qquad \text{for some}\ \alpha _{ 2} \in \mathbb{R},\ \text{conjugate to}\ \alpha _{1}.$$

(42)

Theorem 10.

For any n ≥ 2 there is a constant ν > 0 depending on n only with the following property. For any λ satisfying

$$0 <\lambda \leq \frac{n + 1} {3}$$

(43)

and any interval $J \subset [-\tfrac{1} {2}, \tfrac{1} {2}]$ , for all sufficiently large Q

$$\mu \left (\displaystyle\bigcup _{\alpha _{1}\in \mathbb{A}_{n,\nu }(Q,\lambda )}B(\alpha _{1},{Q}^{-n-1+2\lambda }) \cap J\right ) \geq \tfrac{3} {4}\vert J\vert.$$

(44)

Remark.

The constant $\tfrac{3} {4}$ in the right hand side of (44) can be replaced by any positive number < 1.

Corollary 1.

For any n ≥ 2 there is a positive constant ν depending on n only such that for any λ satisfying (43) and any interval $J \subset [-\tfrac{1} {2}, \tfrac{1} {2}]$ , for all sufficiently large Q

$$\#\big(\mathbb{A}_{n,\nu }(Q,\lambda ) \cap J\big) \geq \tfrac{1} {8}{Q}^{n+1-2\lambda }\vert J\vert.$$

(45)

The deduction of the corollary is rather simple. Indeed, if we have that $B(\alpha _{1},{Q}^{-n-1+2\lambda }) \cap \frac{1} {2}J\not =\varnothing$ then α ₁ ∈ J provided that Q is sufficiently large. Then, using (44) we obtain

$$\displaystyle\begin{array}{rcl} & & \#\big(\mathbb{A}_{n,\nu }(Q,\lambda ) \cap J\big)2{Q}^{-n-1+2\lambda } \geq \\ & &\geq \mu \left (\displaystyle\bigcup _{\alpha _{1}\in \mathbb{A}_{n,\nu }(Q,\lambda )}B(\alpha _{1},{Q}^{-n-1+2\lambda }) \cap \tfrac{1} {2}J\right ){ (44) \atop \geq } \tfrac{1} {4}\vert J\vert , \\ \end{array}$$

whence (45) readily follows. Taking the largest possible value of λ gives Theorem 9.

The key element of the proof of Theorem 10 is a far reaching generalisation of the arguments around (17) shown earlier. The appropriate analogue of Theorem 6 is given by Theorem 5.8 from [7]. In order to give a formal statement we first introduce some further notation. In what follows $\xi _{0},\ldots ,\xi _{n} \in {\mathbb{R}}^{+}$ will be positive real parameters satisfying the following conditions

$$\begin{array}{l} \xi _{i} \ll 1\quad \text{when }0 \leq i \leq m - 1, \\ \xi _{i} \gg 1\quad \text{when }m \leq i \leq n, \\ \xi _{0} < \epsilon ,\qquad \xi _{n} > {\epsilon }^{-1}\end{array}$$

(46)

for some 0 < m ≤ n and $\epsilon > 0$, where the constants in the Vinogradov’s symbol ≪ depend on n only. We will also assume that

$$\displaystyle\prod _{i=0}^{n}\xi _{ i} = 1.$$

(47)

Lemma 3.

For every n ≥ 2 there are positive constants δ ₀ and c₀ depending on n only with the following property. For any interval $J \subset [-\tfrac{1} {2}, \tfrac{1} {2}]$ there is a sufficiently small $\epsilon = \epsilon (n,J) > 0$ such that for any $\xi _{0},\ldots ,\xi _{n}$ satisfying (46) and (47) there is a measurable set G_J ⊂ J satisfying

$$\mu (G_{J}) \geq \tfrac{3} {4}\vert J\vert $$

(48)

such that for every x ∈ G_J there are n + 1 linearly independent primitive irreducible polynomials $P \in \mathbb{Z}[x]$ of degree exactly n such that

$$\delta _{0}\xi _{i} \leq \vert {P}^{(i)}(x)\vert \leq c_{ 0}\xi _{i}\qquad \text{for all }i = 0,\ldots ,n\,.$$

(49)

We now reprocess the main steps of the proof of this statement. Let n ≥ 2 and let $\xi _{0},\ldots ,\xi _{n}$ be given and satisfy (46) and (47) for some m and $\epsilon$. Let $J \subset [-\tfrac{1} {2}, \tfrac{1} {2}]$ be any interval and x ∈ J. Consider the system of inequalities

$$\vert {P}^{(i)}(x)\vert \leq \xi _{ i}\qquad \text{when }0 \leq i \leq n\,,$$

(50)

where $P(x) = a_{n}{x}^{n} +\ldots +a_{1}x + a_{0}$. Let B _x be the set of $(a_{0},\ldots ,a_{n}) \in {\mathbb{R}}^{n+1}$ satisfying (50). Note that B _x is a convex body in ${\mathbb{R}}^{n+1}$ symmetric about the origin. By (47), the volume of B _x equals ${2}^{n+1}\prod _{i=1}^{n}i{!}^{-1}$. Let $\lambda _{0} \leq \lambda _{1} \leq \ldots \leq \lambda _{n}$ be the successive minima of B _x . By Minkowski’s theorem for successive minima,

$$\frac{{2}^{n+1}} {(n + 1)!}\ \leq \ \lambda _{0}\ldots \lambda _{n}\text{vol}B_{x}\ \leq \ {2}^{n+1}.$$

Substituting the value of volB _x gives $\lambda _{0}\ldots \lambda _{n}\ \leq \ \prod _{i=1}^{n}i!$, whence we get that

$$\lambda _{n} \leq \lambda _{0}^{-n}\displaystyle\prod _{ i=1}^{n}i!.$$

(51)

Further we define certain subsets of J that we will ‘avoid’. The avoidance will alow us to find the polynomials P of interest as well as to establish the lower bounds in (49). Let E _∞ (J, δ ₁) be the set of x ∈ J such that λ ₀ = λ ₀(x) ≤ δ ₁, where δ ₁ < 1. By the definition of λ ₀, there is a non-zero polynomial $P \in \mathbb{Z}[x]$, degP ≤ n satisfying

$$\vert {P}^{(i)}(x)\vert \leq \delta _{ 1}\xi _{i}\qquad (0 \leq i \leq n).$$

(52)

Applying Lemma 3 from [7] gives

$$\mu (E_{\infty }(J,\delta _{1})) \ll \left (1 + \frac{1} {\delta _{J}^{\alpha }}\max {\left \{\frac{\delta _{1}\xi _{0}} {\delta _{1}} , \frac{1} {\xi _{n}}\right \}}^{\alpha }\right )\delta _{ 1}^{ \frac{\alpha }{ n+1} }\vert J\vert ,$$

where δ _J  > 0 is a constant. By (46), $\max \{\xi _{0},\xi _{n}^{-1}\} < \epsilon$. Therefore $\mu (E_{\infty }(J,\delta _{1})) \ll \delta _{1}^{\alpha /(n+1)}\vert J\vert$ provided that $\epsilon <\delta _{J}$. Then there is a sufficiently small δ ₁ depending on n only such that

$$\mu (E_{\infty }(J,\delta _{1})) \leq \tfrac{1} {4n+8}\vert J\vert.$$

(53)

By construction, for any $x \in J \setminus E_{\infty }(J,\delta _{1})$ we have that

$$\lambda _{0} \geq \delta _{1}.$$

(54)

Combining (51) and (54) gives

$$\lambda _{n} \leq c_{16} :=\delta _{ 1}^{-n}\displaystyle\prod _{ i=1}^{n}i!,$$

(55)

where c ₁₆ depends on n only. By the definition of λ _n , there are (n + 1) linearly independent integer points $\mathbf{a}_{j} = (a_{0,j},\ldots ,a_{n,j})$ (0 ≤ j ≤ n) lying in the body $\lambda _{n}B_{x} \subset c_{16}B_{x}$. In other words, the polynomials $P_{j}(x) = a_{n,j}{x}^{n} +\ldots +a_{0,j}$ (0 ≤ j ≤ n) satisfy the system of inequalities

$$\vert P_{j}^{(i)}(x)\vert \leq c_{ 16}\xi _{i}\qquad (0 \leq i \leq n).$$

(56)

Let $A = (a_{i,j})_{0\leq i,j\leq n}$ be the integer matrix composed from the integer points a _j (0 ≤ j ≤ n). Since all these points are contained in the body c ₁₆ B _x , we have that | det A | ≪ vol(B _x ) ≪ 1. That is | det A |  < c ₁₇ for some constant c ₁₇ depending on n only. By Bertrand’s postulate, choose a prime number p satisfying

$$c_{17} \leq p \leq 2c_{17}.$$

(57)

Therefore, | det A |  < p. Since $\mathbf{a}_{0},\ldots ,\mathbf{a}_{n}$ are linearly independent and integer, | det A | ≥ 1. Therefore, $\det \,A\not\equiv 0\ ({\rm mod}\ p)$ and the following system

$$A\overline{t} \equiv \overline{b}\ ({\rm mod}\ p)$$

(58)

has a unique non-zero integer solution $\overline{t} {= }^{t}(t_{0},\ldots ,t_{n}) \in {[0,p - 1]}^{n+1}$, where $\overline{b} :{= }^{t}(0,\ldots ,0,1)$ and ^t denotes transposition. For $l = 0,\ldots ,n$ define $\overline{r}_{l} {= }^{t}(1,\ldots ,1,0,\ldots ,0) \in {\mathbb{Z}}^{n+1}$, where the number of zeros is l. Since $\det \,A\not\equiv 0\ ({\rm mod}\ p)$, for every $l = 0,\ldots ,n$ the following system

$$A\overline{\gamma } \equiv -\frac{A\overline{t} -\overline{b}} {p} + \overline{r}_{l}\ ({\rm mod}\ p)$$

(59)

has a unique non-zero integer solution $\overline{\gamma } = \overline{\gamma }_{l} \in {[0,p - 1]}^{n+1}$. Define $\overline{\eta }_{l} := \overline{t} + p\overline{\gamma }_{l}$  (0 ≤ l ≤ n). Consider the (n + 1) polynomials of the form

$$P_{l}(x) = a_{n}{x}^{n} +\ldots +a_{ 0} :=\displaystyle\sum _{ i=0}^{n}\eta _{ l,i}P_{i}(x) \in \mathbb{Z}[x],$$

(60)

where $(\eta _{l,0},\ldots ,\eta _{l,n}) = \overline{\eta }_{l}$. Since $\overline{r}_{0},\ldots ,\overline{r}_{n}$ are linearly independent modulo p, the vectors $-\frac{A\overline{t}-\overline{b}} {p} + \overline{r}_{l}$ ($l = 0,\ldots ,n$) are linearly independent modulo p. Hence, by (59), the vectors $\gamma _{0},\ldots ,\gamma _{n}$ are linearly independent modulo p. Hence, $\gamma _{0},\ldots ,\gamma _{n}$ are linearly independent over $\mathbb{Z}$. Since these vectors are integer, they are also linearly independent over $\mathbb{R}$. Therefore, the vectors $\overline{\eta }_{l} := \overline{t} + p\overline{\gamma }_{l}$  (0 ≤ l ≤ n) are linearly independent over $\mathbb{R}$. Hence the polynomials given by (60) are linearly independent and so are non-zero.

Let $\overline{\eta } = \overline{\eta }_{l}$. Observe that $A\overline{\eta }$ is the column ${}^{t}(a_{0},\ldots ,a_{n})$ of coefficients of P. By construction, $\overline{\eta } \equiv \overline{t}\ ({\rm mod}\ p)$ and therefore $\overline{\eta }$ is also a solution of (58). Then, since $\overline{b} {= }^{t}(0,\ldots ,0,1)$ and $A\overline{\eta } \equiv \overline{b}\ ({\rm mod}\ p)$, we have that $a_{n}\not\equiv 0\ ({\rm mod}\ p)$ and $a_{i} \equiv 0\ ({\rm mod}\ p)$ for $i = 0,\ldots ,n - 1$. Furthermore, by (59), we have that $A\overline{\eta } \equiv \overline{b} + p\overline{r}_{l}\ ({\rm mod}\ {p}^{2})$. Then, on substituting the values of $\overline{b}$ and $\overline{r}_{l}$ into this congruence one readily verifies that $a_{0} \equiv p\ ({\rm mod}\ {p}^{2})$ and so $a_{0}\not\equiv 0\ ({\rm mod}\ {p}^{2})$. By Eisenstein’s criterion, P is irreducible.

Since both $\overline{t}$ and $\overline{\gamma }_{l}$ lie in $\in {[0,p - 1]}^{n+1}$ and $\overline{\eta } = \overline{t} + p\overline{\gamma }_{l}$, it is readily seen that | η _i  | ≤ p ² for all i. Therefore, using (56) and (57) we obtain that

$$\vert {P}^{(i)}(x)\vert \leq c_{ 0}\xi _{i}\qquad (0 \leq i \leq n)$$

(61)

with $c_{0} = 4(n + 1)c_{16}c_{17}^{2}$. Without loss of generality we may assume that the (n + 1) linearly independent polynomials P constructed above are primitive (that is the coefficients of P are coprime) as otherwise the coefficients of P can be divided by their greatest common divisor. Thus, $P \in \mathbb{Z}[x]$ are primitive irreducible polynomials of degree n which satisfy the right hand side of (49). The final part of the proof is aimed at establishing the left hand side of (49).

Let δ ₀ > 0 be a sufficiently small parameter depending on n. For every $j = \overline{0,n}$ let E _j (J, δ ₀) be the set of x ∈ J such that there is a non-zero polynomial $R \in \mathbb{Z}[x]$, degR ≤ n satisfying

$$\vert {R}^{(i)}(x)\vert \leq \delta _{ 0}^{\delta _{i,j} }c_{0}^{1-\delta _{i,j} }\xi _{i},$$

(62)

where δ _i, j equals 1 if i = j and 0 otherwise. Let $\theta _{i} =\delta _{ 0}^{\delta _{i,j}}c_{0}^{1-\delta _{i,j}}\xi _{i}$. Then $E_{j}(J,\delta _{0}) \subset A_{n}(J;\theta _{0},\ldots ,\theta _{n})$. In view of (46) and (47), Lemma 3 from [7] is applicable provided that $\epsilon <\min \{ c_{0}^{-1},c_{0}\delta _{0}\}$. Then, by the same lemma,

$$\mu (E_{j}(J,\delta _{0})) \ll \left (1 + \frac{1} {\delta _{J}^{\alpha }}\max {\left \{ \frac{c_{0}\xi _{0}} {c_{0}^{n}\delta _{0}}, \frac{1} {\delta _{0}c_{0}\xi _{n}}\right \}}^{\alpha }\right ){(\delta _{ 0}c_{0}^{n})}^{1/(n+1)}\vert J\vert \,,$$

where δ _J  > 0 is a constant. It is readily seen that the above maximum is ≤ δ _J if $\epsilon <\delta _{J}\delta _{0}c_{0}$. Then

$$\mu (E_{j}(J,\delta _{0})) \leq \tfrac{1} {4n+8}\vert J\vert $$

(63)

provided that $\epsilon <\min \{\delta _{J}\delta _{0}c_{0},c_{0}^{-1},c_{0}\delta _{0}\}$ and δ ₀ = δ ₀(n) is sufficiently small. By construction, for any x in the set G _J defined by

$$G_{J} := J \setminus \left (\displaystyle\bigcup _{j=0}^{n}E_{ j}(J,\delta _{0}) \cup E_{\infty }(J,\delta _{1})\right )$$

we must necessarily have that | P ⁽ⁱ⁾(x) | ≥ δ ₀ ξ _i for all $i = 0,\ldots ,n$, where P is the same as in (61). Therefore, the left hand side of (49) holds for all i. Finally, observe that

$$\displaystyle\begin{array}{rcl} & & \mu (G_{J}) \geq \vert J\vert -\displaystyle\sum _{i=0}^{n}\mu (E_{ i}(J,\delta _{0})) -\mu (E_{\infty }(J,\delta _{1})) \\ & &{ (53)\,\&\,(63) \atop \geq } \!\vert J\vert - (n + 2)\, \tfrac{1} {(4n+8)}\vert J\vert = \tfrac{3} {4}\vert J\vert.\end{array}$$

The latter verifies (48) and completes the proof.

The following appropriate analogue of Lemma 3 for monic polynomials can be obtained using the techniques of [27].

Lemma 4.

For every n ≥ 2 there are positive constants δ ₀ and c₀ depending on n only with the following property. For any interval $J \subset [-\tfrac{1} {2}, \tfrac{1} {2}]$ there is a sufficiently small $\epsilon = \epsilon (n,J) > 0$ such that for any positive $\xi _{0},\ldots ,\xi _{n}$ satisfying (46) and (47) there is a measurable set G_J ⊂ J satisfying

$$\mu (G_{J}) \geq \tfrac{3} {4}\vert J\vert $$

(64)

such that for every x ∈ G_J there is an irreducible monic polynomials $P \in \mathbb{Z}[x]$ of degree n + 1 satisfying (49).

5 On the Distribution of Resultants

In this section we discuss the distribution of the resultant R(P ₁, P ₂) of polynomials P ₁ and P ₂ from $\mathcal{P}_{n}(Q)$. It is well known that

$$R(P_{1},P_{2}) = a_{n}^{m}b_{ m}^{n}\displaystyle\prod _{ \begin{array}{c}1\leq i\leq n\\ 1\leq j\leq m\end{array}}(\alpha _{i} -\beta _{j}),$$

(65)

where $\alpha _{1},\ldots ,\alpha _{n}$ are the roots of P ₁ and $\beta _{1},\ldots ,\beta _{m}$ are the roots of P ₂; a _n and b _m stand for the leading coefficients of P ₁ and P ₂ respectively, where n = degP ₁ and m = degP ₂. The resultant R(P ₁, P ₂) equals zero if and only if the polynomials P ₁ and P ₂ have a common root. Since the resultant can be represented as the determinant of the Sylvester matrix of the coefficients of P ₁ and P ₂ it follows that R is integer. Furthermore,

$$\vert R(P_{1},P_{2})\vert \ll {Q}^{2n}$$

(66)

for $P_{1},P_{2} \in \mathcal{P}_{n}(Q)$. Akin to the already discussed results for the distribution of determinants we now state their analogue for resultants.

Theorem 11 ([13]). 

Let $m \in \mathbb{Z}$ with 0 ≤ m < n. Then there exist $\gg {Q}^{ \frac{2(n+1)} {(m+1)(m+2)} }$ pairs of different primitive irreducible polynomials (P ₁ ,P ₂ ) from $\mathcal{P}_{n}(Q)$ of degree n such that

$$1 \leq \vert R(P_{1},P_{2})\vert \ll {Q}^{\frac{2(n-m-1)} {m+2} }.$$

(67)

Note that the left had side of (67) is obvious since P ₁ and P ₂ are primitive and irreducible. There are a few interesting corollaries of the above theorem. For m = 0 we have at least c ₁ Q ^n + 1 pairs (P ₁, P ₂) that satisfy $\vert R(P_{1},P_{2})\vert \ll {Q}^{n-1}$. For $m = n - 1$ we have at least $c_{2}{Q}^{\frac{2} {n} }$ pairs (P ₁, P ₂) that satisfy | R(P ₁, P ₂) | ≤ c(n).

To introduce the ideas of the proof we first consider the case m = 0. By Lemma 3 given in the previous section, for any x ∈ G _J there are different irreducible polynomials P ₁ and P ₂ of degree n and height ≪ Q such that

$$\begin{array}{c} \delta _{0}{Q}^{-n} < \vert P_{i}(x_{1})\vert < c_{0}{Q}^{-n},\ i = 1,2 \\ \delta _{0}Q < \vert P{^\prime}_{i}(x_{1})\vert < c_{0}Q, \end{array}$$

(68)

Denote by α ₁ the root of P ₁ closest to x, and by β ₁ the root of P ₂ closest to x. Using (68) and the Mean Value Theorem, one can easily find that

$$\vert x -\alpha _{1}\vert \ll {Q}^{-n-1},\ \vert x -\beta _{ 1}\vert \ll {Q}^{-n-1}.$$

(69)

By (69), we get $\vert \alpha _{1} -\beta _{1}\vert \ll {Q}^{-n-1}$. This together with (65) gives

$$\vert R(P_{1},P_{2})\vert \ll {Q}^{n-1}.$$

(70)

For a fixed pair of (α ₁, β ₁) inequalities (69) are satisfied only for a set of x of measure $\ll {Q}^{-n-1}$. Since μ(G _J ) ≫ | J | , we must have ≫ Q ^n + 1 diffract pairs (α ₁, β ₁) with the above properties. Since each polynomial in $\mathcal{P}_{n}(Q)$ has at most n root, we must have $\gg {Q}^{n+1}$ pairs of different irreducible polynomials (P ₁, P ₂) satisfying (70).

Now let 1 ≤ m ≤ n − 1. Let $v_{0},\ldots ,v_{m} \geq -1$ and $v_{0} + v_{1} +\ldots +v_{m} = n - m$. By Lemma 3, for any x ∈ G _J there exists a pair of irreducible polynomials P ₁, $P_{2} \in \mathbb{Z}[x]$ of degree ≤ n such that for i = 1, 2 we have that

$$\begin{array}{c} \delta _{0}{Q}^{-v_{0}} \leq \vert P_{i}(x)\vert \leq c_{0}{Q}^{-v_{0}},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \, \\ \delta _{0}{Q}^{-v_{j}} \leq \vert P_{i}^{(j)}(x)\vert \leq c_{0}{Q}^{-v_{j}},\ 1 \leq j \leq m, \\ \,\ \ \delta _{0}Q \leq \vert P_{i}^{(j)}(x)\vert \leq c_{0}Q,\ \ \ \ \ m + 1 \leq j.\end{array}$$

(71)

Let $d_{0},d_{1},\ldots ,d_{m+1}$ be a non-increasing sequence of real numbers such that

$$d_{j} = v_{j-1} - v_{j},\ 1 \leq j \leq m,\ d_{m+1} = v_{m} + 1.$$

(72)

Order the roots α _i with respect to x as follows:

$$\vert x -\alpha _{1}\vert \leq \vert x -\alpha _{2}\vert \leq \ldots \leq \vert x -\alpha _{n}\vert.$$

We claim that the roots α _j with 1 ≤ j ≤ m satisfy the following inequalities

$$\begin{array}{c} \vert x -\alpha _{j}\vert \ll {Q}^{-v_{j-1}+v_{j}},\ \ \ \ (1 \leq j \leq m - 1) \\ \vert x -\alpha _{m}\vert \ll {Q}^{-v_{m}-1}.\end{array}$$

(73)

The (j − 1)-th derivative of $P(x) = a_{n}(x -\alpha _{1})\cdots (x -\alpha _{n})$ is

$${P}^{(j-1)}(x) = (j - 1)!a_{ n}\Big(\displaystyle\prod _{i=j}^{n}(x -\alpha _{ i}) +\displaystyle\sum _{i_{j}}(x -\alpha _{i_{1}}) \cdot \ldots \cdot (x -\alpha _{i_{n-j}})\Big),$$

(74)

where the sum $\sum _{i_{j}}$ involves all summands with factor $(x -\alpha _{i_{j}})$, where i _j  < j. If for i < j there is a sufficiently large number s ₁ = c(n) such that

$$\vert x -\alpha _{j}\vert < s_{1}\vert x -\alpha _{i}\vert $$

(75)

then (72) implies (73) for | x − α _j  | . Otherwise, (74) implies

$$\vert {P}^{(j-1)}(x)\vert \gg \vert x -\alpha _{ j}\vert \vert {P}^{(j)}(x)\vert $$

because in this case the summand $(x_{1} -\alpha _{j})(x_{1} -\alpha _{j+1})\cdots (x_{1} -\alpha _{n})$ in the above expression for P ^{(j − 1)}(x ₁) dominates all the others. Now choose v _j so that

$$v_{0} = (m + 1)v_{m} + m\ \text{ and }\ v_{0} = (k + 1)v_{k} - kv_{k+1}\ (1 \leq k \leq m - 1).$$

(76)

By the first equation of (76), we get

$$v_{m} = \frac{v_{0} - m} {m + 1}.$$

(77)

By the other equalities of (76) we have that

$$v_{m-1} = \frac{2v_{0} - m + 1} {m + 1} ,\ \ \ v_{k} = \frac{(m - k + 1)v_{0} - k} {m + 1} \ \ (1 \leq k \leq m - 2).$$

(78)

Finally, by (77) and (78), we obtain

$$v_{j-1} - v_{j} = v_{m} + 1 = \frac{v_{0} + 1} {m + 1}\ \ \ \ (0 \leq j \leq m).$$

(79)

Taking into account the condition

$$v_{0} + v_{1} +\ldots +v_{m} = n - m,$$

by (77) and (78), we have

$$v_{0} = \frac{2n - m} {m + 2}.$$

Thus roots $\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}$ of P ₁ lie within $\ll {Q}^{-(v_{0}+1){(m+1)}^{-1} }$ of x. The same is true for the roots $\beta _{1},\beta _{2},\ldots ,\beta _{m}$ of P ₂. Hence

$$T(m) =\displaystyle\prod _{1\leq i,j\leq m+1}\vert \alpha _{i} -\beta _{j}\vert \ll {Q}^{-\frac{2(n+1)(m+1)} {m+2} }.$$

Consequently

$$\vert R(P_{1},P_{2})\vert \ll {Q}^{\frac{2(n-m-1)} {m+2} }.$$

(80)

It remains to give a lower bound for the number of pairs of (P ₁, P ₂) constructed above. Once again we use the fact that $\alpha _{1},\ldots ,\alpha _{m},\beta _{1},\beta _{m}$ lie within $\ll {Q}^{-(v_{0}+1){(m+1)}^{-1} }$ of x. In other worlds x lies in the interval

$$\Delta (P_{1},P_{2}) =\big\{ x : \vert \max \limits _{i}\{\max \{\alpha _{i} - x\vert ,\vert \beta _{i} - x\vert \}\}\ll {Q}^{-(v_{0}+1){(m+1)}^{-1} }\big\}.$$

Since x is an arbitrary point of G _J and μ(G _J ) ≫ | J | , we must have $\gg {Q}^{(v_{0}+1){(m+1)}^{-1} }$ different pairs (P ₁, P ₂) to cover G _J with intervals $\Delta (P_{1},P_{2})$. Substituting the value of v ₀ we conclude that the number of different pairs (P ₁, P ₂) as above is at least $c(n){Q}^{ \frac{2(n+1)} {(m+1)(m+2)} }$ as required.