One of the most celebrated results in the theory of Sidon sets in the trigonometric system on the circle (or on a compact Abelian group) is Drury’s union theorem that says that the union of two (disjoint) Sidon sets is still a Sidon set. In a recent paper Bourgain and Lewko [2] considered Sidon sets for a general uniformly bounded orthonormal system \((\varphi _n )\) in \(L_2\) over an arbitrary probability space (T, m). They extended some of the classical results known for systems of characters on compact Abelian groups. We continued on the same theme in [6]. Let us recall the basic definitions. We say that \((\varphi _n )\) is Sidon if there is a constant C such that for any finitely supported scalar sequence \(n\mapsto x_n\)

$$\begin{aligned} \sum |x_n| \le C \Vert \sum x_n \varphi _n\Vert _\infty . \end{aligned}$$
(1)

The smallest such C is called the Sidon constant of \((\varphi _n )\). The system \((\varphi _n )\) is called \(\otimes ^k\)-Sidon if the system (\(\varphi _n (t_1 )\varphi _n (t_2 )\cdots \varphi _n (t_k )\)) is Sidon in \(L_2(T^k,m\times \cdots \times m)\). We say that \((\varphi _n )\) is subgaussian if there is a constant \(\beta \) such that for any finite scalar sequence \((x_n)\) such that \(\sum |x_n|^2\le 1\) we have

$$\begin{aligned} \int e^{|\sum x_n \varphi _n|^2/\beta ^2} dm \le e. \end{aligned}$$

When this holds we say that \((\varphi _n )\) is \(\beta \)-subgaussian.

Bourgain and Lewko [2] proved that subgaussian does not imply Sidon but does imply \(\otimes ^5\)-Sidon, and the author [6] improved this to \(\otimes ^2\)-Sidon.

Let \((g_n)\) be an i.i.d. sequence of standard Gaussian random variables. We say that \((\varphi _n )\) is randomly Sidon if there is a constant C such that for any finite scalar sequence \((x_n)\) we have

$$\begin{aligned} \sum |x_n|\le C {\mathbb {E}}\Vert \sum g_n x_n \varphi _n\Vert _\infty . \end{aligned}$$

In [6], we proved that randomly Sidon implies \(\otimes ^4\)-Sidon. It follows as an immediate corollary that the union of two mutually orthogonal Sidon systems is \(\otimes ^4\)-Sidon (see Theorem 15 for a quick outline of a direct proof). This generalizes Drury’s celebrated union theorem for sets of characters. Naturally, this last result raises the question whether \(\otimes ^4\)-Sidon can be replaced by \(\otimes ^k\)-Sidon for \(k<4\). While we cannot decide this for \(k=2\) or \(k=3\), the goal of the present note is to settle the question at least for \(k=1\).

We first improve Bourgain and Lewko’s example from [2] showing that subgaussian does not imply Sidon for uniformly bounded orthonormal systems. Our example is a (very simple) martingale difference sequence and the constant is asymptotically sharp. As a corollary we show that, not surprisingly, Drury’s union theorem does not extend to two mutually orthogonal uniformly bounded orthonormal systems.

FormalPara Theorem 1

Fix \(\varepsilon >0\). There is a uniformly bounded real valued orthonormal system \((\varphi _n)\) with \(\Vert \varphi _n \Vert _\infty \le 1+\varepsilon \) for all n that is subgaussian and actually satisfies

$$\begin{aligned} {\mathbb {E}} e^{ \sum x_n \varphi _n} \le e^{(1+\varepsilon )^2\sum x_n^2/2} \end{aligned}$$
(2)

for any finite sequence of real numbers \((x_n)\), but \((\varphi _n)\) is not a Sidon system.

More precisely, the smallest constant \(C_n\) such that for any scalar coefficients \((x_k)\) we have

$$\begin{aligned} \mathop {\sum }\nolimits _1^n |x_k| \le C_n \Vert \mathop {\sum }\nolimits _1^n x_k \varphi _k\Vert _\infty \end{aligned}$$

satisfies

$$\begin{aligned} \forall n\ge 1\quad C_n\ge \delta _\varepsilon \sqrt{n}, \end{aligned}$$
(3)

where \(\delta _\varepsilon >0\) depends only on \(\varepsilon \). In addition, \((\varphi _n)\) is a martingale difference sequence.

FormalPara Proof

Let \((\varepsilon _n)\) be a sequence of independent choices of signs, i.e. independent ±-valued random variables on a probability space \((\Omega ,{\mathbb {P}})\) taking the values \(\pm 1\) with probabilility 1 / 2. Let \(\mathcal A_n\) be the \(\sigma \)-algebra generated by \(\{\varepsilon _k\mid 0\le k\le n\}\). Let \(0=a_0 \le \cdots \le a_{n-1}\le a_n\le \cdots \) be a fixed non-decreasing sequence for the moment. Consider \(A_0=\Omega \), \(S_0=0\), and define inductively \(A_n\in \mathcal A_n\) and \(S_n\) as follows:

$$\begin{aligned} S_n=S_{n-1}+ \varepsilon _n 1_{A_{n-1}} \text { and } A_{n}=\{ |S_{n}|\le a_{n} \}. \end{aligned}$$

Assume that \({\mathbb {P}}(A_n)\ge \delta \) for some fixed \(\delta >0\). Then let

$$\begin{aligned} f_n= \varepsilon _n 1_{A_{n-1}}. \end{aligned}$$
(4)

This is a martingale difference sequence with \(\Vert f_n\Vert _\infty \le 1\), therefore an orthogonal system such that

$$\begin{aligned} S_n= f_1+\cdots +f_n \end{aligned}$$

and moreover

$$\begin{aligned} \Vert f_n\Vert _2^2 \ge \delta . \end{aligned}$$

We claim that the Sidon constant of \(\{f_1,\ldots ,f_n\}\) is \(\ge n/(1+a_{n-1})\). This follows from the observation that

$$\begin{aligned} \forall n\quad \Vert S_n\Vert _\infty \le 1+a_{n-1}. \end{aligned}$$
(5)

Indeed, this is immediate by induction on n (since either \(\Vert S_n\Vert _\infty \le a_{n-1}+1\) or \(\Vert S_n\Vert _\infty \le \Vert S_{n-1}\Vert _\infty \) depending whether \(\Vert S_n\Vert _\infty \) is attained on \(A_{n-1}\) or on its complement).

Now by Azuma’s inequality (see e.g. [5, p. 501]) we know that \((f_n)\) is subgaussian with a good constant. In fact for any real numbers t and \(x_n\) with \((x_n)\) in \(\ell _2\)

$$\begin{aligned} {\mathbb {E}} e^{t\sum x_n f_n} \le e^{t^2\sum |x_n|^2/2} . \end{aligned}$$
(6)

In particular

$$\begin{aligned} {\mathbb {P}}(\{ |S_n|> t\}) \le 2 e^{-t^2 /2n}. \end{aligned}$$
(7)

Fix \(\varepsilon >0\). Taking \(a_n=c\sqrt{n}\), this gives us

$$\begin{aligned} {\mathbb {P}}(\{ |S_n|> a_n\}) \le 2 e^{-c^2 /2}, \end{aligned}$$

so we can choose a numerical value of c, namely \(c=c_\varepsilon \), large enough so that

$$\begin{aligned} {\mathbb {P}}(\{ |S_n|> a_n\}) \le 1- (1+\varepsilon )^{-2}. \end{aligned}$$

Then we have by what precedes \(\Vert S_n\Vert _\infty \le a_{n-1}+1= c_\varepsilon \sqrt{n-1}+1\) and

$$\begin{aligned} \Vert f_n\Vert _2={\mathbb {P}}(\{ |S_{n-1}|\le a_{n-1} \})^{1/2} \ge (1+\varepsilon )^{-1} \end{aligned}$$

for all n. Therefore the Sidon constant of \(\{f_1,\ldots ,f_n\}\) is \(\ge n/(1+a_{n-1})\). Letting

$$\begin{aligned} \varphi _n=f_n \Vert f_n\Vert _2^{-1} \end{aligned}$$

we find \(\Vert \varphi _n \Vert _\infty \le 1+\varepsilon \) for all n, \( (\varphi _n) \) is orthonormal and (3) holds. By Azuma’s inequality (6) we also have (2). \(\square \)

FormalPara Remark 2

I am grateful to B. Maurey for suggesting the following neater example \((S'_k)\). Let us first fix \(n\ge 1\), and hence \(a_n>0\) is fixed. Let \(M_k=\varepsilon _1+\cdots +\varepsilon _k\) for all \(k\ge 1\). Define the stopping time \(T_n\) by \(T_n=\inf \{k\ge 0\mid |M_k|>a_n\}\) and \(T_n=\infty \) if \(|M_k|\le a_n\) for all \(k\ge 0\). Recall the classical inequalities

$$\begin{aligned} \forall t>0\quad {\mathbb {P}}(\{ \sup _{1\le k\le n} |M_k|> t \} ) \le 2{\mathbb {P}}(\{ |M_n|> t \} )\le 4 e^{-t^2/2n}. \end{aligned}$$

The first one goes back to Paul Lévy (see e.g. [5, p. 28]), it is closely related to Désiré André’s reflection principle for Brownian motion (see e.g. [4, p. 558]) and the second one follows from (7). We then set for \(k\ge 1\)\( S'_k=M_{k\wedge T_n}\) and

$$\begin{aligned} f'_k=S'_k-S'_{k-1}=\varepsilon _k 1_{\{ T_n\ge k \}}. \end{aligned}$$

In the previous example this corresponds to sets \(A'_{k-1}={\{ T_n\ge k \}}={\{ T_n\le k-1 \}^c}\in \mathcal A_{k-1}\). We have clearly \(\Vert S'_k\Vert _\infty \le a_n +1\) for all k, and it is easy to check, since \(A'_{k-1}=\{ \sup _{j<k} |M_j|\le a_n\}\), that we again can choose \(a_n= c_\varepsilon \sqrt{n}\) so that for any \(1\le k\le n\) we have

$$\begin{aligned} {\mathbb {P}}(A'_{k-1})\ge & {} {\mathbb {P}}(A'_{n})={\mathbb {P}}(\{ \sup _{1\le k\le n} |M_k|\le a_n \} )\\= & {} 1-{\mathbb {P}}(\{ \sup _{1\le k\le n} |M_k|> a_n \} ) \ge (1+\varepsilon )^{-2}. \end{aligned}$$
FormalPara Remark 3

Since \((\varphi _n)\) is formed of mean zero variables (2) holds iff there is \(\beta '\) such that

$$\begin{aligned} \forall p\ge 2 \ \forall (x_n)\in \ell _2\quad \Vert \sum x_n \varphi _n\Vert _p\le \beta '\sqrt{p} (\sum |x_n|^2)^{1/2}\end{aligned}$$
(8)
FormalPara Remark 4

Let \((\varphi _n)\) be any orthonormal system. Then for any scalar coefficients \((x_k)\) we have obviously

$$\begin{aligned} \mathop {\sum }\nolimits _1^n |x_k|\le \sqrt{n} (\mathop {\sum }\nolimits _1^n |x_k|^2)^{1/2} \le \sqrt{n} \Vert \mathop {\sum }\nolimits _1^n x_k\varphi _k\Vert _\infty . \end{aligned}$$

Thus the order of growth of the Sidon constant in (3) and the next statement are both sharp.

FormalPara Corollary 5

There are two orthonormal martingale difference sequences \((\varphi ^+_n)\) and \((\varphi ^-_n)\) with orthogonal linear spans such that each has the same distribution as the Rademacher functions (i.e. each is formed of independent ±-valued random variables with mean zero) but their union is not a Sidon system. More precisely the union of \(\{\varphi ^+_k\mid k\le n\}\) and \(\{\varphi ^-_k\mid k\le n\}\) has a Sidon constant \(C_n\) growing like \(\sqrt{n}\).

FormalPara Proof

Let will modify slightly the preceding proof and construct by induction a sequence \(S'_n\). We wish to choose by induction a set \(B_n\subset \Omega \) in \(\mathcal A_n\) (just like \(A_n\) was) and we again set \(S'_n=S'_{n-1}+ \varepsilon _n 1_{B_{n-1}}\). but we choose \(B_n\) satisfying

$$\begin{aligned} B_n \subset \big \{|S'_n|\le a_n\big \} \quad \text { and } {\mathbb {P}}(B_n)=1/2. \end{aligned}$$
(9)

To be able to make this choice all we need to know is that \({\mathbb {P}}(\{|S'_n|\le a_n\}) \ge 1/2.\) Then the preceding argument, associated to \(\varepsilon =\sqrt{2}-1\) still guarantees that \({\mathbb {P}}( \{|S'_{n-1}|\le a_{n-1}\}) \ge 1/2\). Thus we clearly can select \(B_n\) for which (9) holds and we again obtain \(\Vert S'_n\Vert _\infty \le 1+\sqrt{n-1}\) for all n.

Then let

$$\begin{aligned} \varphi ^{\pm }_n= \varepsilon _n ( 1_{B_{n-1}} \pm 1_{\Omega \setminus B_{n-1} }). \end{aligned}$$

Note that since \({\mathbb {P}}(B_{n-1} )=1/2\) we have \( \varphi ^{+}_n \perp \varphi ^{-}_n\) for any n and hence \( \varphi ^{+}_n \perp \varphi ^{-}_k\) for any n, k. Then each of the sequences \(\{\varphi ^{\pm }_k\mid k\le n\}\) is a martingale difference sequence with values in \(\{\pm 1\}\). It is a well known fact (proved by induction as a simple exercise) that this forces each to be distributed uniformly over all choices of signs. Now let \(\{\psi _k\mid k\le 2n\}\) denote the union of the two systems \(\{\varphi ^{+}_k\mid k\le n\}\) and \(\{\varphi ^{-}_k \mid k\le n\}\). Clearly the Sidon constant of \(\{\psi _k\mid k\le 2n\}\) dominates that of \(\{(\varphi ^{+}_k+\varphi ^{-}_k)/2\mid k\le n\}\). But the latter is the system \(\{\varepsilon _k 1_{B_{k-1}} \mid k\le n\}\) as in the preceding proof but with \(B_k\) replacing \(A_k\). Since \(\Vert S'_n \Vert _\infty \le 1+\sqrt{n-1}\), (3) still holds for this system, so the corollary follows. \(\square \)

FormalPara Remark 6

We may clearly replace \((\varepsilon _n)\) by an i.i.d. sequence of complex valued variables \((z_n)\) uniformly distributed over the unit circle of \({\mathbb {C}}\). For those it is still true that for any unimodular sequence \((w_n)\) that is adapted (i.e. \(w_n\) is \(\mathcal A_n\)-measurable for each n) the sequence \((z_n w_{n-1})\) is independent and uniformly distributed over the unit circle. Then the corresponding two sequences \((\varphi ^{\pm }_n)\) are Sidon with constant 1, and their union is not Sidon for the same reason as in the preceding corollary.

Problem: In [2] Bourgain and Lewko show that any n-tuple forming a \(\beta \)-subgaussian orthonormal system uniformly bounded by a constant C contains a subset of cardinality \(\ge \theta n\) with \(\theta =\theta (\beta ,C)>0\) that is Sidon with Sidon constant at most \(f(\beta ,C)\). They ask whether any such system is actually the union of \(k(\beta ,C)\) Sidon sequences with Sidon constant at most \(f(\beta ,C)\).

Is this true for uniformly bounded martingale difference sequences normalized in \(L_2\) ?

Although for the example appearing in the proof of Theorem 1 the answer is affirmative (consider e.g. a partition into odd and even k’s), we believe that a more involved one with values in \(\{-1,0,1\}\) as in (4) but with a more subtle choice of the predictable sets \(A_{n-1}\), should yield a counterexample.

Let \(M_n\) be the space of \(n\times n\)-matrices with complex entries, equipped with the usual operator norm on the n-dimensional Hilbert space. In [6] we consider a non-commutative analogue involving a \(n\times n\)-matrix valued function \(\varphi (t)=[\varphi (t)_{ij}]\) on a probability space (T, m) for which the uniform boundedness condition is replaced by

$$\begin{aligned} \Vert \varphi (t)\Vert _{M_n} \le C \end{aligned}$$

and we assume that \(\{\sqrt{n} \varphi (t)_{ij}\mid 1\le i,j\le n\}\) is \(\beta \)-subgaussian and orthonormal. The prototypical example is when \(\varphi \) is uniformly distributed over the unitary group.

In this situation we prove in [6, Prop. 5.4] that there is a constant \(\alpha =\alpha (C,\beta )\) such that

$$\begin{aligned} \forall a\in M_n \quad \mathrm{tr}|a|\le \alpha \sup _{t_1,t_2\in T} |\mathrm{tr}(a\varphi (t_1)\varphi (t_2))| . \end{aligned}$$

In analogy with Theorem 1 it is natural to wonder what is the best constant \(C'_n \) such that in the same situation

$$\begin{aligned} \forall a\in M_n \quad \mathrm{tr}|a|\le C'_n \sup _{t \in T} |\mathrm{tr}(a\varphi (t) )| . \end{aligned}$$

Clearly the orthonormality assumption yields

$$\begin{aligned} \forall a\in M_n \quad n^{-1} \mathrm{tr}|a|\le & {} (n^{-1} \mathrm{tr}|a|^2)^{1/2}= \Vert \mathrm{tr}(a\varphi (t ) \Vert _2\le \Vert \mathrm{tr}(a\varphi (t ) \Vert _\infty \\= & {} \sup _{t \in T} |\mathrm{tr}(a\varphi (t)|. \end{aligned}$$

and hence \(C'_n \le n\).

It is easy to see that this is asymptotically optimal. Indeed, consider the following example. Let \(x\mapsto D(x)\) be the mapping taking an \(n\times n\) matrix to its diagonal part. Let u denote a random \(n\times n\) unitary matrix uniformly distributed over the unitary group. Let \((\varphi _1 , \ldots ,\varphi _n )\) be the orthonormal n-tuple constructed in the proof of Theorem 1, of which we keep the notation, namely \(\varphi _k=f_k \Vert f_k\Vert _2^{-1}\). Assuming (T, m) large enough, we define \(\varphi : T \rightarrow M_n\) so that \(\varphi -D(\varphi )\) and \(D(\varphi )\) are independent random variables; we make sure that \(\varphi -D(\varphi )\) and \(u -D(u)\) have the same distribution and we adjust the diagonal entries of \(D(\varphi )\) so that they have the same distribution as \((\varphi _1/\sqrt{n}, \ldots ,\varphi _n/\sqrt{n})\). Then for a suitable \(\beta \) (independent of n) \(\{\sqrt{n} \varphi (t)_{ij}\mid 1\le i,j\le n\}\) is \(\beta \)-subgaussian and orthonormal. However, if a is the diagonal matrix with entries \((\Vert f_1\Vert _2, \ldots ,\Vert f_n\Vert _2)\) we have on one hand by (5) \(\Vert \mathrm{tr}(a\varphi )\Vert _\infty = \Vert (f_1 + \cdots +f_n)/\sqrt{n}\Vert _\infty \le c_\varepsilon \), and on the other hand \(\mathrm{tr}|a|\ge n(1+\varepsilon )^{-1}\). Therefore

$$\begin{aligned} C'_n\ge n(1+\varepsilon )^{-1} c_\varepsilon ^{-1}. \end{aligned}$$
FormalPara Definition 7

Let I be an index set. Let \(L_1(m'),L_1(m'')\) be arbitrary \(L_1\)-spaces. We say that a family \((f_n)_{n\in I}\) in \(L_1(m'')\) is c-dominated by another one \((\psi _n)_{n\in I}\) in \(L_1(m')\) if there is a linear map \(u:\ L_1(m')\rightarrow L_1(m'')\) with \(\Vert u\Vert \le c\) such that \(u(\psi _n)= f_n\) for all \(n\in I\).

The following criterion due to M. Lévy (see [3] and [6, Prop.1.5]) is very useful: a linear map \(v: E \rightarrow L_1(m'')\) on a subspace \(E\subset L_1(m')\) admits an extension \( u: L_1(m') \rightarrow L_1(m'')\) with \(\Vert u\Vert \le 1\) iff for any finite sequence \((\eta _n)\) in E we have

$$\begin{aligned} \Vert \sup |v(\eta _n)|\Vert _{L_1(m'')} \le \Vert \sup |\eta _n|\Vert _{L_1(m')}. \end{aligned}$$
(10)

If we apply this to \(E=\mathrm{span}[\psi _n]\) with v defined by \(v(\psi _n)= f_n\), this gives us the following criterion: a sequence \((f_n)_{n\in I}\) in \({L_1(m'')}\) is c-dominated by a sequence \((\psi _n)_{n\in I}\) in \({L_1(m')}\) iff for any Banach space B and any finite sequence \((x_n)\) in B we have

$$\begin{aligned} \Vert \sum f_n x_n \Vert _{L_1(B)} \le c \Vert \sum \psi _n x_n \Vert _{L_1(B)} . \end{aligned}$$
(11)

Indeed, it is easy to see that we may restrict consideration to the single space \(B=\ell _\infty \), in which case (10) and (11) are identical.

FormalPara Remark 8

The key fact used in [6] is that, for some numerical constant K, any \(\beta \)-subgaussian sequence \((\varphi _n)_{n\in {\mathbb {N}}}\) in \(X=L_1(T,m)\) is \(K\beta \)-dominated by a standard i.i.d. sequence of Gaussian normal variables (on a probability space \((\Omega ',{\mathbb {P}}')\)), denoted by \((g_n)_{n\in {\mathbb {N}}}\). This is essentially due to Talagrand; see [6] for detailed references and comments. It would be interesting to have a direct simple proof of this fact.

If we assume moreover that the \(\beta \)-subgaussian sequence \((\varphi _n)_{n\in {\mathbb {N}}}\) is uniformly bounded, i.e. that \(\Vert \varphi _n\Vert _\infty \le \alpha \) for all n, then, for some numerical constant \(K'\), the sequence \((\varphi _n)_{n\in {\mathbb {N}}}\) is \(K'(\beta + \alpha )\)-dominated by \((\varepsilon _n)\). This follows from the solution by Bednorz and Latała [1] of Talagrand’s Bernoulli conjecture.

We would like to observe that if \((f_n)\) is a martingale difference sequence then a very simple proof is available (with an optimal constant). We start with a special case of the form \(f_n=\varepsilon _n \varphi _{n-1}\) with \(\varphi _{n-1}\) depending only on \(\varepsilon _1,\ldots ,\varepsilon _{n-1}\) satisfying \(\Vert \varphi _{n-1}\Vert _\infty \le 1\) (which is subgaussian by (6)). This is particularly easy. Indeed, for any \(y\in [-1,1]\) let

$$\begin{aligned} F(t,y)=(-1) 1_{[0,(1-y)/2]}(t) + (1)1_{((1-y)/2,1]}(t), \end{aligned}$$

so that \(\int _0^1 F(t,y) dt= y\) and \( F(t,y) =\pm 1\). Let us consider the sequence of random variables \(F_n\) defined on \([0,1]^{\mathbb {N}}\times \{-1,1\}^{\mathbb {N}}\) by setting

$$\begin{aligned} F_n( (t_j), (\varepsilon _j)) = \varepsilon _n F(t_{n-1}, \varphi _{n-1}). \end{aligned}$$

Let u be the conditional expectation onto the algebra of functions depending on the second variable on \([0,1]^{\mathbb {N}}\times \{-1,1\}^{\mathbb {N}}\). Then \(u(F_n)=f_n\). Moreover since \((F_n)\) is a martingale with values in \(\pm 1\) it has the same distribution as \((\varepsilon _n)\) itself. In other words, there is an isometry \(v :L_1(\Omega ,{\mathbb {P}}) \rightarrow L_1([0,1]^{\mathbb {N}}\times \{-1,1\}^{\mathbb {N}})\) such that \(v(\varepsilon _n)= F_n\) for all n. Considering the composition uv, this shows that \((f_n)\) is 1-dominated by \((\varepsilon _n)\), and the latter is easily shown to be c-dominated by \((g_n)\) (the latter being, say, in \(L_1(\Omega ',{\mathbb {P}}')\)) for some numerical constant c.

More generally, let \((\Omega ', \mathcal A',{\mathbb {P}}')\) be an arbitrary probability space. We have

FormalPara Lemma 9

Let \(\varphi \in L_1(\Omega ', \mathcal A',{\mathbb {P}}')\) be with values in \([-1,1]\) and such that \({\mathbb {E}} \varphi =0\). Then for any Banach space B and any \(x_0,x_1\in B\)

$$\begin{aligned} {\mathbb {E}}'\Vert x_0+\varphi x_1\Vert \le {\mathbb {E}}\Vert x_0+\varepsilon _1 x_1\Vert . \end{aligned}$$
(12)

More generally, if \(\mathcal B\subset \mathcal A'\) is any \(\sigma \)-subalgebra such that \({\mathbb {E}}^{\mathcal B}\varphi =0\) we have for any \(x_0\in L_1(\Omega ', \mathcal B,{\mathbb {P}}'; B)\)

$$\begin{aligned} {\mathbb {E}}'\Vert x_0+\varphi x_1\Vert \le {\mathbb {E}}'{\mathbb {E}}\Vert x_0+\varepsilon _1 x_1\Vert . \end{aligned}$$
(13)
FormalPara Proof

We have

$$\begin{aligned} x_0+\varphi x_1= \int x_0 + F(t, \varphi ) x_1 dt. \end{aligned}$$

and hence by Jensen

$$\begin{aligned} \Vert x_0+\varphi x_1\Vert\le & {} \int \Vert x_0 + F(t, \varphi ) x_1 \Vert dt=\Vert x_0 - x_1 \Vert (1-\varphi )/2 \\&+\Vert x_0 + x_1 \Vert (1+\varphi )/2. \end{aligned}$$

After integration, we obtain (12). To prove (13) it suffices to show that

$$\begin{aligned} {\mathbb {E}}^{\mathcal B}\Vert x_0+\varphi x_1\Vert \le {\mathbb {E}}^{\mathcal B} (\Vert x_0+ x_1\Vert +\Vert x_0- x_1\Vert )/2, \end{aligned}$$
(14)

or equivalently that for any \(A\in \mathcal B\) with \({\mathbb {P}}'(A)>0\) we have

$$\begin{aligned} {\mathbb {P}}'(A)^{-1}\int _A\Vert x_0+\varphi x_1\Vert d{\mathbb {P}}'\le {\mathbb {P}}'(A)^{-1}\int _A (\Vert x_0+ x_1\Vert +\Vert x_0- x_1\Vert )/2 d{\mathbb {P}}', \end{aligned}$$
(15)

Assume that \(A\in \mathcal B\) is an atom of \(\mathcal B\). Then \(x_0\) is constant on A and \({\mathbb {E}}^{\mathcal B}\) when restricted to A coincides with the average over A. Thus (15) reduces to (13) with \({\mathbb {P}}'\) replaced by \({\mathbb {P}}'(A)^{-1} {\mathbb {P}}'_{|A}\). The case of a general \(A\in \mathcal B\) can be proved by a routine approximation argument left to the reader. \(\square \)

We now show that any real valued martingale difference sequence with values in \([-1,1]\) is 1-dominated by \((\varepsilon _n)\).

FormalPara Lemma 10

Let \((d_n)\) be a sequence of real valued martingale differences on \((\Omega ', \mathcal A',{\mathbb {P}}')\), i.e. there are \(\sigma \)-subalgebras \(\mathcal A_n\subset \mathcal A\) (\(n\ge 0\)) forming an increasing filtration such that \(d_n\) is \({\mathcal A}_n\)-measurable for all \(n\ge 0\) and \({\mathbb {E}}^{{\mathcal A}_{n-1} }d_n=0\) for all \(n\ge 1 \). We assume that \(\mathcal A_0\) is trivial (so that \(d_0\) is constant). If \( |d_n|\le 1\) a.s. for any n, then there is an operator \(u: L_1(\Omega , \mathcal A,{\mathbb {P}}) \rightarrow L_1(\Omega ', \mathcal A',{\mathbb {P}}')\) with \(\Vert u\Vert = 1\) such that \(u(1)=1\) and \(u(\varepsilon _n)=d_n\) for all \(n\ge 1 \).

FormalPara Proof

By the above criterion (11) it suffices to show that for any Banach space B and any finite sequence \((x_n)\) in B we have for any k

$$\begin{aligned} \Vert d_0 x_0+ \mathop {\sum }\nolimits _1^{k} d_n x_n\Vert _{L_1(B)} \le \Vert d_0 x_0+ \mathop {\sum }\nolimits _1^{k} \varepsilon _n x_n\Vert _{L_1(B)}. \end{aligned}$$
(16)

By (13) with \(\mathcal B= \mathcal A_{k-1}\) and \(\varphi =d_k\) we have

$$\begin{aligned} \Vert d_0 x_0+ \mathop {\sum }\nolimits _1^{k} d_n x_n\Vert _{L_1(B)} \le \Vert d_0 x_0+ \mathop {\sum }\nolimits _1^{k-1} d_n x_n + \varepsilon _k x_k\Vert _{L_1({\mathbb {P}}'\times {\mathbb {P}};B)} . \end{aligned}$$

Now working on the product space \((\Omega , \mathcal A,{\mathbb {P}}) \times (\Omega ', \mathcal A',{\mathbb {P}}')\) with \(\mathcal B\) equal to \(\sigma ( \mathcal A_{k-2} \cup \varepsilon _k) \) we find

$$\begin{aligned}&\Vert d_0 x_0+ \mathop {\sum }\nolimits _1^{k-1} d_n x_n + \varepsilon _k x_k\Vert _{L_1({\mathbb {P}}'\times {\mathbb {P}};B)} \le \Vert d_0 x_0\\&\quad +\mathop {\sum }\nolimits _1^{k-2} d_n x_n + \varepsilon _{k-1} x_{k-1} + \varepsilon _k x_k\Vert _{L_1({\mathbb {P}}'\times {\mathbb {P}};B)}. \end{aligned}$$

Continuing in this way we obtain (16). \(\square \)

FormalPara Remark 11

(On the complex valued case in Lemma 10) Let \({\mathbb {T}}={\mathbb {R}}/2\pi {\mathbb {Z}}\) be the (one dimensional) torus. Consider the sequence \((z_n)_{n\in {\mathbb {N}}}\) formed of the coordinate functions on \({\mathbb {T}}^{{\mathbb {N}}} \) equipped with its normalized Haar measure \(\mu \). A priori the complex analogue of the preceding proof, with \((z_n)\) replacing \((\varepsilon _n)\), requires to assume that the martingale under consideration is a Hardy martingale in the sense described e.g. in [5, p. 133]. Indeed, the Poisson kernel is the natural analogue of the barycentric argument we use for Lemma 9. Using this, Lemma 10 remains valid, with \((z_n)\) replacing \((\varepsilon _n)\), for a martingale difference sequence \((d_n)\) adapted to the usual filtration on \({\mathbb {T}}^{\mathbb {N}}\) such that for any n the variable \(z\mapsto d_n(z_0,\ldots ,z_{n-1}, z)\) is either analytic or anti-analytic.

Note that without any additional assumption the complex valued case of Lemma 10 fails, simply because the system \((1,\varepsilon _1)\) is not 1-dominated by \((1,z_1)\). Indeed, by (10) this would imply the inequality \(2=\int \max \{|1+\varepsilon _1| ,|1-\varepsilon _1| \}d{\mathbb {P}}\le \int \max \{|1+z_1| ,|1-z_1| \}d\mu \), which clearly fails.

The next two remarks will be used at the very end of this paper.

FormalPara Remark 12

Let \((z_n)_{n\in {\mathbb {N}}}\) and \(\mu \) on \({\mathbb {T}}^{{\mathbb {N}}} \) be as in Remark 11. Consider two sequences \((f^1_n)\) and \((f^2_n)\) in an \(L_1\)-space X. We form their “disjoint union” \((f_n)\) by setting \(f_{2k}= f^2_k\) and \(f_{2k+1}= f^1_k\). We claim that if \((f^1_n)\) (resp. \((f^2_n)\)) is \(c_1\)-dominated (resp. \(c_2\)-dominated) by \((z_n)\), then \((f_n)\) is \((c_1+c_2)\)-dominated by \((z_n)\). Actually, the same claim is valid for the disjoint union of arbitrary families indexed by sets \(I_1\) and \(I_2\) (using \((z_n)_{n\in {I_1 \dot{\cup }I_2}}\) on \({\mathbb {T}}^{I_1 \dot{\cup }I_2}\) instead), but the idea is easier to describe with \(I={\mathbb {N}}\). Indeed, since \((z_n)\), \((z_{2n})\) and \((z_{2n+1})\) all have the same distribution, there is \(u_j:\ L_1({\mathbb {T}}^{\mathbb {N}},\mu )\rightarrow X\) (\(j=1,2\)) with \(\Vert u_j\Vert \le c_j\) such that \(u_2(z_{2n})=f^2_n\) and \(u_1(z_{2n+1})=f^1_n\). Let \({\mathbb {E}}_1\) and \({\mathbb {E}}_2\) be the conditional expectations on \(L_1({\mathbb {T}}^{\mathbb {N}},\mu )\) with respect to the \(\sigma \)-algebras generated respectively by \((z_{2n+1})\) and \((z_{2n})\). Then let \(u=u_1{\mathbb {E}}_1+u_2{\mathbb {E}}_2\). We have \(u(z_n)=f_n\) for all n and \(\Vert u\Vert \le \Vert u_1{\mathbb {E}}_1\Vert +\Vert u_2{\mathbb {E}}_2\Vert \le c_1+c_2\). This proves our claim.

FormalPara Remark 13

Let \((z_n)\) be as in Remark 12 on \(({\mathbb {T}}^{{\mathbb {N}}},\mu )\). Let \((\varphi _n)\) be in \(L_\infty (T,m)\). We claim that if \(\Vert \varphi _n\Vert _\infty \le 1\) for all n, then \((\varphi _n \otimes z_n)\) is dominated by \((z_n)\). Assume first \(|\varphi _n|=1\) a.e. for all n. Then the translation invariance of the distribution of \((z_n)\) shows that \((\varphi _n \otimes z_n)\) has the same distribution as \((z_n)\), so the claim is obvious in this case. Note that any number \(\varphi \in {\mathbb {C}}\) with \(|\varphi |\le 1\) is an average of two points on the unit circle. Using this it is easy to verify the claim. It can also be checked easily using the criterion in (11).

We end this paper by an outline of a proof that the union of two Sidon sequences is \(\otimes ^4\)-Sidon, more direct than the one in [6]. The route we use avoids the consideration of randomly Sidon sequences, it is essentially the commutative analogue of the proof in [7], with the free Abelian group replacing the free group. The key fact for the latter route is still the following:

FormalPara Lemma 14

Let \((z_n)\) be in \(L_\infty ({\mathbb {T}}^{{\mathbb {N}}} ,\mu )\) as in Remark 12. Let (T, m) be a probability space. Let \((f_n)\) be a sequence in \(L_1(T,m)\) that is dominated by \((z_n)\). Then any sequence \((\psi _n)\) in \(L_\infty (T,m)\) that is both uniformly bounded and biorthogonal to \((f_n)\) is \(\otimes ^2\)-Sidon. Here biorthogonal means

$$\begin{aligned} \forall n,m \quad \int \psi _n f_m= \delta _{nm}. \end{aligned}$$
FormalPara Proof

Let \(u: L_1({\mathbb {T}}^{{\mathbb {N}}} ,\mu ) \rightarrow L_1(T,m)\) such that \(u(z_n)=f_n\). Elementary considerations show that it suffices to show that the sequence \((u^*(\psi _n))\) is \(\otimes ^2\)-Sidon. By another elementary argument \((u^*(\psi _n))\) is biorthogonal to \((z_n)\). Therefore, it suffices to prove this Lemma for the case \((T,m) =({\mathbb {T}}^{{\mathbb {N}}} ,\mu )\) and \((\psi _n)=(z_n)\). This is proved in [6] with \((z_n)\) replaced by an i.i.d. gaussian sequence, using the Ornstein-Uhlenbeck (or Mehler) semigroup. Here we may use Riesz products instead.

We claim that for any N and any \(z^0\in {\mathbb {T}}^{{\mathbb {N}}}\) the function \(F=\mathop {\sum }\nolimits _1^N z_n^0 z_n \otimes z_n \) admits for any \(0<\varepsilon \le 1\) a decomposition \(F=t_\varepsilon +r_\varepsilon \) in the algebraic tensor product \(L_1({\mathbb {T}}^{{\mathbb {N}}})\otimes L_1({\mathbb {T}}^{{\mathbb {N}}})\) with

$$\begin{aligned} \Vert t_\varepsilon \Vert _{\wedge }=\int |t_\varepsilon (x,y)| d\mu (x)d\mu (y) \le w(\varepsilon )\text { and } \Vert r_\varepsilon \Vert _{\vee } \le \varepsilon , \end{aligned}$$

where we have set

$$\begin{aligned} \Vert r_\varepsilon \Vert _{\vee } =\sup \nolimits _{a,b\in B_{L_\infty }} \left| \int r_\varepsilon (x,y) a(x)b(y) d\mu (x)d\mu (y)\right| , \end{aligned}$$

and where \(w(\varepsilon )\) is a function depending only on \(0<\varepsilon \le 1\) (and not on N or \(z^0\)). To verify this we fix \(z^0\) and consider in \(L_1( {\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}, \mu \times \mu )\) the Riesz product

$$\begin{aligned} \nu _\varepsilon (z,z')= \prod \nolimits _1^N \big (1+\varepsilon \mathfrak {R}\big (z_n^0 z_nz_n'\big )\big )=\mathop {\sum }\limits _{\alpha \subset [1...N] } (\varepsilon /2)^{|\alpha |} \prod _{n\in \alpha } \big (z_n^0 z_nz_n'+\overline{z_n^0 z_nz_n'}\big ). \end{aligned}$$

We will view the tensors in \(L_1({\mathbb {T}}^{{\mathbb {N}}})\otimes L_1({\mathbb {T}}^{{\mathbb {N}}})\) as functions of \((z,z')\in {\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}\). Note

$$\begin{aligned} \nu _\varepsilon (z,z') =\mathop {\sum }\nolimits _{\alpha \subset [1...N] } (\varepsilon /2)^{|\alpha |} \sum _{\beta \subset \alpha }\ \prod _{n\in \beta } z_n^0 z_nz_n' \prod _{n\in [1...N]\setminus \beta } \overline{z_n^0 z_nz_n'}. \end{aligned}$$
(17)

Observe that the terms of the latter sum are orthogonal. Without trying to optimize (see [6] for a discussion of the optimal logarithmic growth for w) we set

$$\begin{aligned} t'_\varepsilon = ( \nu _\varepsilon -\nu _0)/\varepsilon . \end{aligned}$$

Note that (since \(\nu _\varepsilon \ge 0\) and hence \(\Vert \nu _\varepsilon \Vert _1=1\)) we have \(\Vert t'_\varepsilon \Vert _{\wedge }\le 2/\varepsilon \). Let \(r'_\varepsilon = \mathop {\sum }\nolimits _1^N \mathfrak {R}(z_n^0 z_nz_n')) -t'_\varepsilon \).

By the orthogonality in the sum (17) one checks that \(\Vert r'_\varepsilon \Vert _{\vee }\le \varepsilon /2\). This gives us the desired decomposition but, instead of \(\mathop {\sum }\nolimits _1^N {z_n^0 z_nz_n'}\), we are decomposing the sum

$$\begin{aligned} \mathop {\sum }\nolimits _1^N \mathfrak {R}\big (z_n^0 z_nz_n'\big )\big )=(1/2)\mathop {\sum }\nolimits _1^N {z_n^0 z_nz_n'} + (1/2)\mathop {\sum }\nolimits _1^N \overline{z_n^0 z_nz_n'} . \end{aligned}$$

To remove the second term we introduce an extra variable \(\omega \in {\mathbb {T}}\) that acts on \({\mathbb {T}}^{\mathbb {N}}\) by multiplication ( i.e. \(\omega (z_n)= (\omega z_n) \)) and we define (here \(m_{\mathbb {T}}\) is normalized Haar measure on \({\mathbb {T}}\))

$$\begin{aligned} t_\varepsilon (z,z')=2\int \bar{\omega } t'_\varepsilon (\omega z,z') dm_{\mathbb {T}}(\omega ) \text { and } r_\varepsilon (z,z')=2\int \bar{\omega } r'_\varepsilon (\omega z,z') dm_{\mathbb {T}}(\omega ). \end{aligned}$$

This gives us \(\Vert t_\varepsilon \Vert _{\wedge }\le 4/\varepsilon \) and \(\Vert r_\varepsilon \Vert _{\vee }\le \varepsilon \). Moreover we have

$$\begin{aligned} (1/2)\mathop {\sum }\nolimits _1^N {z_n^0 z_nz_n'}= (1/2) t_\varepsilon + (1/2) r_\varepsilon \end{aligned}$$

which proves the claim with \(w(\varepsilon )\le 4/\varepsilon \).

We can now complete the proof. Let \((a_n)\) be a scalar sequence. Let \(\Psi = \mathop {\sum }\nolimits _1^N a_n \psi _n \otimes \psi _n\). Choosing \(z_n^0\) so that \(z_n^0 a_n= |a_n|\) we have

$$\begin{aligned} \langle \Psi ,F \rangle =\sum z_n^0 a_n=\sum |a_n|, \end{aligned}$$

and hence \(\sum |a_n|=\langle \Psi , t_\varepsilon \rangle + \langle \Psi , r_\varepsilon \rangle \) which leads to

$$\begin{aligned} \sum |a_n| \le \Vert \Psi \Vert _\infty w(\varepsilon ) +\sum |a_n| \varepsilon \big (\sup \nolimits _{1\le n\le N} \Vert \psi _n\Vert _\infty ^2\big ). \end{aligned}$$

To conclude, we set \(C'=\sup \nolimits _{n\ge 1} \Vert \psi _n\Vert _\infty \) and we choose, say, \(\varepsilon = 1/2C'^2\). We have then

$$\begin{aligned} \sum |a_n| \le 2 w(\varepsilon ) \Vert \Psi \Vert _\infty . \end{aligned}$$

\(\square \)

Let us say that a bounded set S in \( L_\infty (T,m)\) is Sidon with constant C if for any finitely supported function \(x: S \rightarrow {\mathbb {C}}\) we have \(\mathop {\sum }\nolimits _{\varphi \in S} |x(\varphi )| \le C \Vert \sum x(\varphi ) \varphi \Vert .\) If \((\varphi _n)\) is an enumeration of S, this is the same as \(\mathop {\sum }\nolimits _{n\in {\mathbb {N}}} |x(n)| \le C \Vert \mathop {\sum }\nolimits _{n\in {\mathbb {N}}} x(n) \varphi _n\Vert .\) Similarly we extend the term \(\otimes ^4\)-Sidon to sets in \( L_\infty (T,m)\).

For the convenience of the reader we give a slightly more direct proof of the following result from [6], which generalizes Drury’s theorem.

FormalPara Theorem 15

Let \(\Lambda _1=\{\varphi ^1_n\mid n\in I(2)\}\) and \( \Lambda _2=\{\varphi ^2_n\mid n\in I(1)\}\) be two Sidon sets (indexed by sets I(1), I(2)) in \( L_\infty (T,m)\), with constants \(C_1,C_2\). Assume that \( \Lambda _1 \perp \Lambda _2\) in \(L_2(m)\) and there are \(C'_1,C'_2,\delta >0 \) such that

$$\begin{aligned} \forall n \quad \delta \le \Vert \varphi ^1_n\Vert _2\le \Vert \varphi ^1_n\Vert _\infty \le C'_1 \text { and } \delta \le \Vert \varphi ^2_n\Vert _2\le \Vert \varphi ^2_n\Vert _\infty \le C'_2 . \end{aligned}$$

Then the union \( \Lambda _1 \cup \Lambda _2\) is \(\otimes ^4\)-Sidon with a constant C depending only on \(C_1,C_2,C'_1,C'_2,\delta \).

FormalPara Proof

We assume for simplicity that the sets are sequences indexed by \({\mathbb {N}}\). By homogeneity (changing \(C'_1,C'_2\) accordingly) we may assume that \(\Vert \varphi ^1_n\Vert _2=\Vert \varphi ^2_n\Vert _2=1\) for all n. Let \(E_j\subset L_\infty (T,m)\) be the norm closed span of \((\varphi ^j_n)\) (\(j=1,2\)). Consider the linear mapping \(T_j: E_j \rightarrow L_\infty ({\mathbb {T}}^{{\mathbb {N}}})\) such that \(T_j(\varphi ^j_n )= z_n\). By assumption \(\Vert T_j\Vert \le C_j\). By the injectivity of \(L_\infty \)-spaces \(T_j\) has an extension \(\widetilde{T}_j: L_\infty (T,m) \rightarrow L_\infty ({\mathbb {T}}^{{\mathbb {N}}})\) such that \(\widetilde{T_j}_{| E_j} =T_j\) and \(\Vert \widetilde{T}_j \Vert =\Vert T_j\Vert \le C_j\). We introduce the operator \( {\mathcal T} : L_\infty (T,m) \rightarrow L_\infty ({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}})\) defined by

$$\begin{aligned} {\mathcal T} ( f)(z,z')= \widetilde{T_1}(f) (z) + \widetilde{T_2}(f) (z'). \end{aligned}$$

Then \(\Vert {\mathcal T} \Vert \le C_1+C_2\). The operator \( {\mathcal T} \otimes id_{L_\infty (T,m)} \) clearly extends to an bounded operator

$$\begin{aligned} W: L_\infty (T\times T) \rightarrow L_\infty ({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}\times T), \end{aligned}$$

satisfying \(\Vert W\Vert \le \Vert {\mathcal T} \Vert \le C_1+C_2\).

We claim that the collection

$$\begin{aligned} \mathcal U= \big \{W\big (\varphi _n^1\otimes \varphi _n^1\big )\big \} \cup \big \{W\big (\varphi _n^2\otimes \varphi _n^2\big )\big \} \end{aligned}$$

is biorthogonal to

$$\begin{aligned} \mathcal V= \{\overline{z_n\otimes 1\otimes \varphi _n^1}\} \cup \{\overline{1\otimes z_n\otimes \varphi _n^2}\}. \end{aligned}$$

Indeed, note \(W(\varphi _n^1\otimes \varphi _n^1) \subset L_\infty ({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}) \otimes \varphi _n^1\) and \(W(\varphi _n^2\otimes \varphi _n^2) \subset L_\infty ({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}) \otimes \varphi _n^2\). Therefore, by our \(L_2(m)\)-orthogonality assumption

$$\begin{aligned} \forall n,m\quad W\big (\varphi _n^1\otimes \varphi _n^1\big ) \perp {1\otimes z_m\otimes \varphi _m^2}\text { and } W\big (\varphi _n^2\otimes \varphi _n^2\big ) \perp { z_m\otimes 1\otimes \varphi _m^2}. \end{aligned}$$

Moreover, if we set \(\xi _n^1= \widetilde{T_2}(\varphi _n^1) \) we have

$$\begin{aligned} W\big (\varphi _n^1\otimes \varphi _n^1\big ) = {\mathcal T}\big (\varphi _n^1\big )\otimes \varphi _n^1 = z_n \otimes 1 \otimes \varphi _n^1 + 1\otimes \xi _n^1 \otimes \varphi _n^1, \end{aligned}$$

which shows that \((W(\varphi _n^1\otimes \varphi _n^1))\) is biorthogonal to \( \{\overline{z_n\otimes 1\otimes \varphi _n^1}\}\). Similarly \((W(\varphi _n^2\otimes \varphi _n^2))\) is biorthogonal to \( \{\overline{ 1\otimes z_n\otimes \varphi _n^2}\}\). This proves the claim.

By Remarks 13 and 12, the family \( \mathcal V =\{\overline{z_n\otimes 1\otimes \varphi _n^1}\} \cup \{\overline{1\otimes z_n\otimes \varphi _n^2}\} \) is dominated in \(L_1({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}\times T)\) by the sequence \((z_n)\). By Lemma 14 we conclude that \(\mathcal U\) is \(\otimes ^2\)-Sidon in \(L_\infty ({\mathbb {T}}^{{\mathbb {N}}}\times {\mathbb {T}}^{{\mathbb {N}}}\times T) \). Since W is bounded this implies that \(\{\varphi _n^1\otimes \varphi _n^1\} \cup \{\varphi _n^2\otimes \varphi _n^2\} \) is also \(\otimes ^2\)-Sidon in \(L_\infty (T\times T) \). Consequently \( \Lambda _1 \cup \Lambda _2\) is \(\otimes ^4\)-Sidon in \(L_\infty (T,m)\). The assertion about the constant C is easy to check by going over the various steps. \(\square \)