Abstract
We prove that for almost every (1,3) configuration, there is no linear dependence between the associated time-frequency translates of any \(f\in L^2(\mathbb {R})\backslash \{0\}\).
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1 Introduction
For a measurable function \(f:\mathbb {R}\rightarrow \mathbb {C}\) and a subset \(\Lambda \subset \mathbb {R}^2\), the associated Gabor system is given by
where
We call \(M_bT_af\) a time-frequency translate of f.
The Heil–Ramanathan–Topiwala (HRT) conjecture [10] asserts that finite Gabor systems in \(L^2(\mathbb {R})\) are linearly independent (also see [8]). That is
The HRT Conjecture Let \(\Lambda \subset \mathbb {R}^2\) be a finite set. Then there is no non-trivial function \( f \in L^2(\mathbb {R})\) such that the associated Gabor system \( {\mathcal {G}}(f,\Lambda )\) is linearly dependent in \(L^2(\mathbb {R})\).
Here are some examples to show that the \(L^2(\mathbb {R})\) property of function f is essential.
-
1
For any trigonometric polynomial f, there exists a finite subset \(\Lambda \subset \mathbb {R}^2\) such that the associated Gabor system \( {\mathcal {G}}(f,\Lambda )\) is linearly dependent.
-
2
Let \(f(x)=\frac{1}{2^n}\) for \(x\in [n-1,n)\). Then \(f\in L^2(\mathbb {R}^+)\) but \(f\notin L^2(\mathbb {R})\). It is easy to see that \(\{f(x+1),f(x)\}\) is linearly dependent.
Since the formulation of the HRT conjecture, some results were obtained (see [9] and references therein) under further restrictions on the behavior of function f(x) at \(x=\infty \) [1, 3, 4, 10] or the structure of the time-frequency translates \(\Lambda \) [2, 5, 6, 10, 12]. Recall that we call \(\Lambda \) an (n, m) configuration if there exist 2 distinct parallel lines containing \(\Lambda \) such that one of them contains exactly n points of \(\Lambda \), and the other one contains exactly m points of \(\Lambda \). The following results hold without restriction on \(f\in L^2(\mathbb {R})\backslash \{0\}\).
-
\( {\mathcal {G}}(f,\Lambda )\) is linearly independent if \(\# \Lambda \le 3\) or \(\Lambda \) is colinear [10].
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\( {\mathcal {G}}(f,\Lambda )\) is linearly independent if \(\Lambda \) is a finite subset of a translate of a lattice in \(\mathbb {R}^2\) [12]. See [2, 6] for alternative proofs.
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\( {\mathcal {G}}(f,\Lambda )\) is linearly independent if \(\Lambda \) is a (2,2) configuration [5, 7].
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\( {\mathcal {G}}(f,\Lambda )\) is linearly independent if \(\Lambda \) is a (1,3) configuration with certain arithmetic restriction [5]. See Theorem 1.1.
In this paper, we consider (1, 3) configurations. In [5], Demeter proved
Theorem 1.1
The HRT conjecture holds for special (1, 3) configurations
-
(a)
if there exists \(\gamma >1\) such that
$$\begin{aligned} \liminf _{n\rightarrow \infty }n^{\gamma }\min \Big \{\Big \Vert n\frac{\beta }{\alpha }\Big \Vert , \Big \Vert n\frac{\alpha }{\beta }\Big \Vert \Big \}<\infty , \end{aligned}$$(1) -
(b)
if at least one of \(\alpha ,\beta \) is rational.
It is known that \(\{x\in \mathbb {R}: \text { there exists some } \gamma >1 \text { such that } \liminf _{n\rightarrow \infty }n^{\gamma } ||n x||<\infty \}\) is a set of zero Lebesgue measure (e.g., [11, Theorem 32]). Thus Theorem 1.1 holds for a measure zero subset of parameters, and it has been an open problem to extend it to other (1, 3) configurations.
Our main result is
Theorem 1.2
The HRT conjecture holds for special (1, 3) configurations
-
(a)
if
$$\begin{aligned} \liminf _{n\rightarrow \infty }n\ln n\min \Big \{\Big \Vert n\frac{\beta }{\alpha }\Big \Vert , \Big \Vert n\frac{\alpha }{\beta }\Big \Vert \Big \}<\infty , \end{aligned}$$(2) -
(b)
if at least one of \(\alpha ,\beta \) is rational.
Remark 1.3
(b) of Theorem 1.2 is the same statement as (b) in Theorem 1.1. We list here for completeness.
It is well known that \(\{x\in \mathbb {R}: \liminf _{n\rightarrow \infty }n\ln n ||n x||<\infty \}\) is a set of full Lebesgue measure (e.g., [11, Theorem 32]). Then using metaplectic transformations, we have the following theorem
Theorem 1.4
Given any line l in \(\mathbb {R}^2\), let \((a_2,b_2)\) and \((a_3,b_3)\) be any two points lying in l. Let \((a_1,b_1)\) be an any point not lying in l. Then for almost every point \((a_4,b_4)\) in l, the HRT conjecture holds for the configuration \(\Lambda =\{(a_1,b_1),(a_2,b_2),(a_3,b_3),(a_4,b_4)\}\).
Proof
By metaplectic transformations (see [10] for details), we can assume l is x-axis, and \((a_1,b_1)=(0,1)\), \((a_2,b_2)=(0,0)\) and \((a_3,b_3)=(\alpha ,0)\). By Theorem 1.2 and the fact that \(\{x\in \mathbb {R}: \liminf _{n\rightarrow \infty }n\ln n ||n x||<\infty \}\) is a set of full Lebesgue measure, we have for almost every \(\beta \), the HRT conjecture holds for \(\Lambda =\{(0,1),(0,0),(\alpha ,0),(\beta ,0)\}\). We finish the proof. \(\square \)
2 The Framework of the Proof of Theorem 1.2
If \(\frac{\alpha }{\beta }\) is a rational number, it reduces to the lattice case, which has been proved [12]. Thus we also assume \(\frac{\alpha }{\beta }\) is irrational.
Assume Theorem 1.2 does not hold. Then there exists some function f satisfying
and nonzero \(C_i\in \mathbb {C}\) such that
(Theorem 1.2 is covered by the known results if \(C_i=0\) for some \(i=0,1,2\))
Let
For \(n>0\), define
Notice that \(P(x+n)\) is an almost periodic function. Thus for almost every \(x\in [0,1)\),
Iterating (4) n times on both sides (positive and negative), we have for \(n>0\),
and
This implies that the value of function f on \(\mathbb {R}\) can be determined uniquely by its value on [0, 1) and function P(x).
By Egoroff’s theorem and conditions (3), (5) and (6), there exists some positive Lebesgue measure set \(S\subset [0,1)\) and \(d>0\), such that
and
Demeter constructed a sequence \(\{n_k\}\subset \mathbb {Z}^+\), such that
for some \(x_k,x^\prime _k\in S\). This contradicts (7)–(10).
In order to complete the construction of (11), growth condition (1) was necessary in [5]. In the present paper, we follow the approach of [5]. The novelty of our work is in the subtler Diophantine analysis. This allows to make the restriction weak enough to obtain the result for a full Lebesgue measure set of parameters, and significantly simplifies the proof.
The rest of the paper is organized as follows. In Sect. 3, we will give some basic facts. In Sect. 4, we give the proof of Theorem 1.2.
3 Preliminaries
We start with some basic notations. Denote by [x], \(\{x\}\), \(\Vert x\Vert \) the integer part, the fractional part and the distance to the nearest integer of x. Let \(\langle x\rangle \) be the unique number in \([-1/2,1/2)\) such that \(x-\langle x\rangle \) is an integer. For a measurable set \(A\subset \mathbb {R}\), denote by |A| its Lebesgue measure.
For any irrational number \(\alpha \in \mathbb {R}\), we define
and inductively for \(k>0,\)
We define
and inductively,
Recall that \(\{q_n\}_{n\in \mathbb {N}}\) is the sequence of denominators of best approximations of irrational number \(\alpha \), since it satisfies
Moreover, we also have the following estimate,
Lemma 3.1
Let \( k_1< k_2< k_3<\cdots <k_m\) be a monotone integer sequence such that \(k_m-k_1< q_n\). Suppose for some \(x\in \mathbb {R}\)
Then
Proof
Recall that \(\langle x\rangle \) is the unique number in \([-1/2,1/2)\) such that \(x-\langle x\rangle \) is an integer. Thus \(||x||=|\langle x\rangle |\). In order to prove the Lemma, it suffices to show that
Let \(S^+=\{j: j=1,2,\cdots ,m , \langle k_j\alpha -x\rangle >0\}\). Let \(j_0^+\) be such that \(j_0^+\in S^+\), and
By (14) and (15), one has for \( i\ne j \) and \( i,j\in S^+\),
It implies the gap between any two points \( \langle k_i\alpha -x \rangle \) and \(\langle k_j\alpha -x \rangle \) with \( i,j\in S^+\) is larger than \(\frac{1}{2q_n}\). See the following figure.
It easy to see that the upper bound of \( \sum _{j\in S^+}\frac{1}{||k_j\alpha -x||}\) is achieved if all the gaps are exactly \( \frac{1}{2q_n}\). In this case, the gap between the ith closest points of \(\langle k_j\alpha -x \rangle \) with \(j\in S^+\) to \( \langle k_{j_0^+}\alpha -x \rangle \) is exactly \( \frac{i}{2q_n}\). Thus by (16), we have
Similarly, letting \(S^-=\{j: j=1,2,\cdots ,m , \langle k_j\alpha -x\rangle <0\}\), one has
By (19) and (20), we finish the proof. \(\square \)
Now we give two lemmas which can be found in [5].
Lemma 3.2
([5, Lemma 2.1]) Let \(C_0,C_1,C_2\in \mathbb {C}\backslash \{0\}\). The polynomial \(p(x,y)=C_0+C_1e^{2\pi i x}+C_2e^{2\pi i y}\) has at most two real zeros \((\gamma _1^{(j)},\gamma _2^{(j)})\in [0,1)^2\), \(j\in \{1,2\}\) and there exists \(t=t(C_0,C_1,C_2)\in \mathbb {R}\setminus \{0\}\) such that
for each \(x,y\in \mathbb {R}\).
Remark 3.3
In (21), we assume p(x, y) has two zeros. If p(x, y) has one or no zeros, we can proceed with our proof by replacing (21) with
or
Lemma 3.4
([5, Lemma 4.1]) Let \(x_1,x_2,\ldots ,x_N\) be N not necessarily distinct real numbers. Then for each \(N\in \mathbb {Z}^+\) and each \(\delta >0\), there exists a set \(E_{N,\delta }\subset [0,1)\) with \(|E_{N,\delta }|\le \delta ,\) such that
and
for each \(x\in [0,1)\setminus E_{N,\delta }\).
4 Proof of Theorems 1.2
In this section, \(q_k,p_k,a_k\) are always the coefficients of the continued fraction expansion of \(\frac{\alpha }{\beta }\) as given in (12) and (13). Then condition (2) holds iff
and also iff
Lemma 4.1
Suppose \(\frac{\alpha }{\beta }\) is irrational and satisfies condition (2). Then for any \(s\in (0,1)\), there exists a sequence \(N_k\) such that
-
(i)
$$\begin{aligned} N_k=m_{n_k}q_{n_k}, m_{n_k}\le C(s), \end{aligned}$$(25)
-
(ii)
$$\begin{aligned} \Big \Vert N_k\frac{\alpha }{\beta }\Big \Vert \le \frac{C(s)}{N_k\ln N_k}, \end{aligned}$$(26)
and
-
(iii)
$$\begin{aligned} \Big \{\frac{N_k}{\beta }\Big \} \le s. \end{aligned}$$(27)
Proof
By (24), there exists a sequence \({n_k}\) such that \( a_{n_k}\ge c \ln q_{n_k}\). For any \(s\in (0,1)\), let \(m_{n_k}\in \mathbb {Z}^+\) be such that \(1\le m_{n_k}\le 1/s+1\) and \(N_k =m_{n_k}q_{n_k}\) satisfies (iii) (this can be done by the pigeonhole principle). It is easy to check that \( N_k\) satisfies condition (ii) by the fact \( a_{n_k}\ge c \ln q_{n_k}\). \(\square \)
Lemma 4.2
Let \(C_0,C_1,C_2\in \mathbb {C}\backslash \{0\}\) and \(\alpha ,\beta \) be such that \(\frac{\alpha }{\beta }\) is irrational. Let \(Q_k\) be a sequence such that \(\gamma q_{n_k}\le Q_k\le {\hat{\gamma }} q_{n_k}\), where \(q_n\) is the continued fraction expansion of \(\frac{\alpha }{\beta }\) and \(\gamma ,{\hat{\gamma }}\) are constants. Define
Then for each \(\delta >0\), there exists a set \(E_{Q_k,\delta }\subset [0,1)\) such that
and
for each \(x\in [0,1)\setminus E_{Q_k,\delta }\).
Proof
In order to make the proof simpler, we will use C for constants depending on \(\gamma ,{\hat{\gamma }},\delta ,C_0,C_1,C_2,\alpha ,\beta \).
Let \((\gamma _1,\gamma _2)\) be a zero of the polynomial \(p(x,y)=C_0+C_1e^{2\pi i x}+C_2e^{2\pi iy}\), and let t be the real number given by Lemma 3.2. Define
By Lemma 3.2, it suffices to find a set with \(|E_{Q_k,\delta }|\le \delta ,\) such that
for each \(x\in [0,1)\setminus E_{Q_k,\delta }\).
We distinguish between two cases.
Case 1 \(\alpha +t\beta \not =0\)
In this case, one has
where \(m=-1\) if \(\{\beta (x+n)-\gamma _2\}>1/2\) and \(m=0\) otherwise. We remind that \([\beta (x+n)-\gamma _2]\) is the integer part of \(\beta (x+n)-\gamma _2\).
Note that the set
has \(O(Q_k)\) elements. By (22) there exists some \(E^1_{Q_k,\delta }\) with \(|E^1_{Q_k,\delta }|<\delta /2\) such that
for each \(x\in [0,1)\setminus E^1_{Q_k,\delta }\). This implies (28).
Case 2\(\alpha +t\beta =0\).
In this case, one has
where m is as before. Let \(\xi \) be either \(\gamma _1+t\gamma _2\) or \(\gamma _1+t\gamma _2+t\), depending on whether \(m=0\) or \(-1\). From Lemma 3.1, we have that for each \(x\in [0,1)\)
Let \(S(\xi )\) (not depending on x) be the set of those \(0\le n\le Q_k-1\) such that \(\Vert \frac{\alpha }{\beta }[\beta (x+n)-\gamma _2]-\xi \Vert \le \frac{1}{4q_{n_k}}\) for some \(x\in [0,1)\). It is easy to see that \(\# S(\xi )\le C\) by (14) and (15). For \(n\in S(\xi )\), we will use an alternative estimate
By (23), there exists some set \(E^{2}_{Q_k,\delta }\subset [0,1)\) such that \(|E^{2}_{Q_k,\delta }|\le \frac{\delta }{2}\) and
for each \(x\in [0,1)\setminus E^{2}_{Q_k,\delta }\). Thus in this case, (28) follows from (29) and (30). Putting two cases together, we finish the proof. \(\square \)
Theorem 4.3
Under the conditions of Lemma 4.2, let \(N_k\) be a sequence such that (i), (ii) and (iii) in Lemma 4.1 hold. Define \(P_k:=\frac{N_k}{\beta }\) for \(\beta >0\) and \(P_k:=-\frac{N_k}{\beta }\) for \(\beta <0\). Given \(\delta >0\), there exists \(E_{k,\delta }\subset [0,1)\) with \(|E_{k,\delta }|\le \delta \) such that for each x, y satisfying \(x\in [0,1)\setminus E_{k,\delta }\) and \(x=y+P_k\), we have
Proof
We write C for \(C(\delta ,s,C_0,C_1,C_2,\alpha ,\beta )\) again. Without loss of generality, we only consider the case \(\beta >0\).
By (26) we have
and
Thus, for each \(n\in \mathbb {Z}^+\), one has
By the fact \(1+x\le e^{x}\) for \(x>0\), we get
and thus
Now Theorem 4.3 follows from Lemma 4.2. \(\square \)
Proof of Theorem 1.2
Suppose Theorem 1.2 is not true. As argued in Sect. 2, there there exist some function f, a positive Lebesgue measure set \(S\subset [0,1)\) and \(d>0\) such that (7)–(10) hold. By the continuity of Lebesgue measure, there exists \(\varepsilon =\varepsilon (S)>0\) such that
for \(\{P_k\}\le \varepsilon \). Let \(\delta =\frac{|S|}{100}\). Then \((S\setminus E_{k,\delta })\cap (\{P_k\}+ S)\not =\emptyset \). Let \(s=\varepsilon \). Applying Theorem 4.3 with s and \(\delta \), we have
for each \(x\in [0,1)\setminus E_{k,\delta }\) and \(x=y+P_k\).
Now we can choose \(x_k\in S\setminus E_{k,\delta }\) such that \(x_k-\{P_k\}\in S\). Let \(y_k=x_k'-[P_k]=x_k-P_k\). Then
Applying (9) and (10) with \(x_k,x^\prime _k\in S\), one has
and
By (8), (33) and (34), we obtain that
This is contradicted by (7), if we let \(k\rightarrow \infty \). \(\square \)
References
Benedetto, J.J., Bourouihiya, A.: Linear independence of finite Gabor systems determined by behavior at infinity. J. Geom. Anal. 25(1), 226–254 (2015)
Bownik, M., Speegle, D.: Linear independence of Parseval wavelets. Ill. J. Math. 54(2), 771–785 (2010)
Bownik, M., Speegle, D.: Linear independence of time-frequency translates of functions with faster than exponential decay. Bull. Lond. Math. Soc. 45(3), 554–566 (2013)
Bownik, M., Speegle, D.: Linear independence of time-frequency translates in \({\mathbb{R}}^d\). J. Geom. Anal. 26(3), 1678–1692 (2016)
Demeter, C.: Linear independence of time frequency translates for special configurations. Math. Res. Lett. 17(4), 761–779 (2010)
Demeter, C., Gautam, S.Z.: On the finite linear independence of lattice Gabor systems. Proc. Am. Math. Soc. 141(5), 1735–1747 (2013)
Demeter, C., Zaharescu, A.: Proof of the HRT conjecture for \((2,2)\) configurations. J. Math. Anal. Appl. 388(1), 151–159 (2012)
Heil, C.: Linear independence of finite Gabor systems. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 171–206. Springer, New York (2006)
Heil, C., Speegle, D.: The HRT conjecture and the zero divisor conjecture for the Heisenberg group. In: Balan, R., et al. (eds.) Excursions in Harmonic Analysis. Appl. Numer. Harmon. Anal., vol. 3, pp. 159–176. Birkhäuser/Springer, Cham (2015)
Heil, C., Ramanathan, J., Topiwala, P.: Linear independence of time-frequency translates. Proc. Am. Math. Soc. 124(9), 2787–2795 (1996)
Khinchin, A.Y.: Continued Fractions. Dover Publications, Inc., Mineola (1997)
Linnell, P.A.: von Neumann algebras and linear independence of translates. Proc. Am. Math. Soc. 127(11), 3269–3277 (1999)
Acknowledgements
I would like to thank Svetlana Jitomirskaya for introducing to me the HRT conjecture and inspiring discussions on this subject. The author was supported by the AMS-Simons Travel Grant 2016–2018 and NSF DMS-1700314. This research was also partially supported by NSF DMS-1401204.
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Liu, W. Letter to the Editor: Proof of the HRT Conjecture for Almost Every (1,3) Configuration. J Fourier Anal Appl 25, 1350–1360 (2019). https://doi.org/10.1007/s00041-018-9628-0
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DOI: https://doi.org/10.1007/s00041-018-9628-0