A note on Sidon sets in bounded orthonormal systems

TL;DR

This paper constructs explicit examples of orthonormal systems with bounded elements that are Sidon only with large constants, demonstrating optimal bounds and exploring their structural properties in harmonic analysis.

Contribution

It provides the first explicit examples of bounded orthonormal systems that are Sidon only with constants proportional to \,\sqrt{n}\,, and analyzes their structural and domination properties.

Findings

Constructed explicit examples with optimal Sidon constants.

Showed union of two Sidon sequences can have large Sidon constant.

Proved martingale differences are dominated by Rademacher sequences.

Abstract

We give a simple example of an $n$-tuple of orthonormal elements in $L_2$ (actually martingale differences) bounded by a fixed constant, and hence subgaussian with a fixed constant but that are Sidon only with constant $\approx \sqrt n$. This is optimal. The first example of this kind was given by Bourgain and Lewko, but with constant $\approx \sqrt {\log n}$. We also include the analogous $n\times n$-matrix valued example, for which the optimal constant is $\approx n$. We deduce from our example that there are two $n$-tuples each Sidon with constant 1, lying in orthogonal linear subspaces and such that their union is Sidon only with constant $\approx \sqrt n$. This is again asymptotically optimal. We show that any martingale difference sequence with values in $[-1,1]$ is "dominated" in a natural sense (related to our results) by any sequence of independent, identically distributed, symmetric $\{-1,1\}$-valued variables (e.g. the Rademacher functions). We include a self-contained proof that any sequence $(\varphi_n)$ that is the union of two Sidon sequences lying in orthogonal subspaces is such that $(\varphi_n\otimes\varphi_n \otimes\varphi_n\otimes\varphi_n)$ is Sidon.

Equations

$$\sum |x_n|\le C\Vert \sum x_n{\phi }_n{\Vert }_{\infty }.$$

$$\int e^{|\sum x_n{\phi }_n{|}^2/{\beta }^2}dm\le e.$$

$$\sum |x_n|\le C\mathbb{E}\Vert \sum g_n{x}_n{\phi }_n{\Vert }_{\infty }.$$

$$\mathbb{E}{e}^{\sum x_n{\phi }_n}\le e^{(1+\epsilon {)}^2\sum x_n^2/2}$$

$${\sum }_1^n|x_k|\le C_n\Vert {\sum }_1^n{x}_k{\phi }_k{\Vert }_{\infty }$$

$$\forall n\ge 1C_n\ge {\delta }_{\epsilon }\sqrt{n},$$

$$S_n=S_{n-1}+{\epsilon }_n{1}_{A_{n-1}}\text{ and }{A}_n=\{|S_n|\le a_n\}.$$

$$f_n={\epsilon }_n{1}_{A_{n-1}}.$$

$$S_n=f_1+\cdots +f_n$$

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

$$\forall n\Vert S_n{\Vert }_{\infty }\le 1+a_{n-1}.$$

$$\mathbb{E}{e}^{t\sum x_n{f}_n}\le e^{t^2\sum |x_n{|}^2/2}.$$

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

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

$$\mathbb{P}(\{|S_n|>a_n\})\le 1-(1+\epsilon {)}^{-2}.$$

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

$${\phi }_n=f_n\Vert f_n{\Vert }_2^{-1}$$

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

$$f_k^{\prime }=S_k^{\prime }-S_{k-1}^{\prime }={\epsilon }_k{1}_{\{T_n\ge k\}}.$$

$$\mathbb{P}(A_{k-1}^{\prime })\ge \mathbb{P}(A_n^{\prime })=\mathbb{P}(\{\underset{1\le k\le n}{\sup }|M_k|\le a_n\})=1-\mathbb{P}(\{\underset{1\le k\le n}{\sup }|M_k|>a_n\})\ge (1+\epsilon {)}^{-2}.$$

$$\forall p\ge 2\forall (x_n)\in {\ell }_2\Vert \sum x_n{\phi }_n{\Vert }_p\le {\beta }^{\prime }\sqrt{p}(\sum |x_n{|}^2{)}^{1/2}$$

$${\sum }_1^n|x_k|\le \sqrt{n}({\sum }_1^n|x_k{|}^2{)}^{1/2}\le \sqrt{n}\Vert {\sum }_1^n{x}_k{\phi }_k{\Vert }_{\infty }.$$

$$B_n\subset \{|S_n^{\prime }|\le a_n\}\text{ and }\mathbb{P}(B_n)=1/2.$$

$${\phi }_n^{\pm }={\epsilon }_n(1_{B_{n-1}}\pm 1_{\Omega \setminus B_{n-1}}).$$

$$\Vert \phi (t){\Vert }_{M_n}\le C$$

$$\forall a\in M_n\mathrm{tr}|a|\le \alpha \underset{t_1,t_2\in T}{\sup }|\mathrm{tr}(a\phi (t_1)\phi (t_2))|.$$

$$\forall a\in M_n\mathrm{tr}|a|\le C_n^{\prime }\underset{t\in T}{\sup }|\mathrm{tr}(a\phi (t))|.$$

$$\forall a\in M_n{n}^{-1}\mathrm{tr}|a|\le (n^{-1}\mathrm{tr}|a{|}^2{)}^{1/2}=\Vert \mathrm{tr}(a\phi (t){\Vert }_2\le \Vert \mathrm{tr}(a\phi (t){\Vert }_{\infty }=\underset{t\in T}{\sup }|\mathrm{tr}(a\phi (t)|.$$

$$C_n^{\prime }\ge n(1+\epsilon {)}^{-1}{c}_{\epsilon }^{-1}.$$

$$\Vert \sup |v({\eta }_n)|{\Vert }_{L_1(m^{\prime \prime })}\le \Vert \sup |{\eta }_n|{\Vert }_{L_1(m^{\prime })}.$$

Full text

MSC: 43A46,42C05 (primary). 60642, 60646 (secondary).

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 $n\mapsto x_{n}$

[TABLE]

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 $\sum|x_{n}|^{2}\leq 1$ we have

[TABLE]

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

[TABLE]

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.

[TABLE]

*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*

[TABLE]

satisfies

[TABLE]

where $\delta_{\varepsilon}>0$ depends only on $\varepsilon$. In addition, $(\varphi_{n})$ is a martingale difference sequence.

[TABLE]

Assume that $\mathbb{P}(A_{n})\geq\delta$ for some fixed $\delta>0$. Then let

[TABLE]

This is a martingale difference sequence with $\|f_{n}\|_{\infty}\leq 1$, therefore an orthogonal system such that

[TABLE]

and moreover

[TABLE]

We claim that the Sidon constant of $\{f_{1},\cdots,f_{n}\}$ is $\geq n/(1+a_{n-1})$. This follows from the observation that

[TABLE]

Indeed, this is immediate by induction on $n$ (since either $\|S_{n}\|_{\infty}\leq a_{n-1}+1$ or $\|S_{n}\|_{\infty}\leq\|S_{n-1}\|_{\infty}$ depending whether $\|S_{n}\|_{\infty}$ 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 $\ell_{2}$

[TABLE]

In particular

[TABLE]

Fix $\varepsilon>0$. Taking $a_{n}=c\sqrt{n}$, this gives us

[TABLE]

so we can choose a numerical value of $c$, namely $c=c_{\varepsilon}$, large enough so that

[TABLE]

Then we have by what precedes $\|S_{n}\|_{\infty}\leq a_{n-1}+1=c_{\varepsilon}\sqrt{n-1}+1$ and

[TABLE]

for all $n$. Therefore the Sidon constant of $\{f_{1},\cdots,f_{n}\}$ is $\geq n/(1+a_{n-1})$. Letting

[TABLE]

we find $\|\varphi_{n}\|_{\infty}\leq 1+\varepsilon$ for all $n$, $(\varphi_{n})$ is orthonormal and (3) holds. By Azuma’s inequality (6) we also have (2). ∎

[TABLE]

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 $k\geq 1$ $S^{\prime}_{k}=M_{k\wedge T_{n}}$ and

[TABLE]

In the previous example this corresponds to sets $A^{\prime}_{k-1}={\{T_{n}\geq k\}}={\{T_{n}\leq k-1\}^{c}}\in\mathcal{A}_{k-1}$. We have clearly $\|S^{\prime}_{k}\|_{\infty}\leq a_{n}+1$ for all $k$, and it is easy to check, since $A^{\prime}_{k-1}=\{\sup_{j<k}|M_{j}|\leq a_{n}\}$, that we again can choose $a_{n}=c_{\varepsilon}\sqrt{n}$ so that for any $1\leq k\leq n$ we have

[TABLE]

[TABLE]

[TABLE]

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

[TABLE]

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

Then let

[TABLE]

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 $\{\varphi^{\pm}_{k}\mid k\leq n\}$ 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 $\{\psi_{k}\mid k\leq 2n\}$ denote the union of the two systems $\{\varphi^{+}_{k}\mid k\leq n\}$ and $\{\varphi^{-}_{k}\mid k\leq n\}$. Clearly the Sidon constant of $\{\psi_{k}\mid k\leq 2n\}$ dominates that of $\{(\varphi^{+}_{k}+\varphi^{-}_{k})/2\mid k\leq n\}$. But the latter is the system $\{\varepsilon_{k}1_{B_{k-1}}\mid k\leq n\}$ as in the preceding proof but with $B_{k}$ replacing $A_{k}$. Since $\|S^{\prime}_{n}\|_{\infty}\leq 1+\sqrt{n-1}$, (3) still holds for this system, so the corollary follows. ∎

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 $\geq\theta n$ 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

[TABLE]

and we assume that $\{\sqrt{n}\varphi(t)_{ij}\mid 1\leq i,j\leq n\}$ 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

[TABLE]

In analogy with Theorem 1 it is natural to wonder what is the best constant $C^{\prime}_{n}$ such that in the same situation

[TABLE]

Clearly the orthonormality assumption yields

[TABLE]

and hence $C^{\prime}_{n}\leq n$.

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 $(\varphi_{1},\cdots,\varphi_{n})$ be the orthonormal $n$-tuple constructed in the proof of Theorem 1, of which we keep the notation, namely $\varphi_{k}=f_{k}\|f_{k}\|_{2}^{-1}$. Assuming $(T,m)$ large enough, we define $\varphi:T\to M_{n}$ 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 $(\varphi_{1}/\sqrt{n},\cdots,\varphi_{n}/\sqrt{n})$. Then for a suitable $\beta$ (independent of $n$) $\{\sqrt{n}\varphi(t)_{ij}\mid 1\leq i,j\leq n\}$ is $\beta$-subgaussian and orthonormal. However, if $a$ is the diagonal matrix with entries $(\|f_{1}\|_{2},\cdots,\|f_{n}\|_{2})$ we have on one hand by (5) $\|{\rm tr}(a\varphi)\|_{\infty}=\|(f_{1}+\cdots+f_{n})/\sqrt{n}\|_{\infty}\leq c_{\varepsilon}$, and on the other hand ${\rm tr}|a|\geq n(1+\varepsilon)^{-1}$. Therefore

[TABLE]

The following criterion due to M. Lévy (see [3] and [6, Prop.1.5]) is very useful: a linear map $v:E\to L_{1}(m^{\prime\prime})$ on a subspace $E\subset L_{1}(m^{\prime})$ admits an extension $u:L_{1}(m^{\prime})\to L_{1}(m^{\prime\prime})$ with $\|u\|\leq 1$ iff for any finite sequence $(\eta_{n})$ in $E$ we have

[TABLE]

If we apply this to $E={\rm span}[\psi_{n}]$ with $v$ defined by $v(\psi_{n})=f_{n}$, this gives us the following criterion: a sequence $(f_{n})_{n\in I}$ in ${L_{1}(m^{\prime\prime})}$ is $c$-dominated by a sequence $(\psi_{n})_{n\in I}$ in ${L_{1}(m^{\prime})}$ iff for any Banach space $B$ and any finite sequence $(x_{n})$ in $B$ we have

[TABLE]

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.

If we assume moreover that the $\beta$-subgaussian sequence $(\varphi_{n})_{n\in\mathbb{N}}$ is uniformly bounded, i.e. that $\|\varphi_{n}\|_{\infty}\leq\alpha$ for all $n$, then, for some numerical constant $K^{\prime}$, the sequence $(\varphi_{n})_{n\in\mathbb{N}}$ is $K^{\prime}(\beta+\alpha)$-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 $\varepsilon_{1},\cdots,\varepsilon_{n-1}$ satisfying $\|\varphi_{n-1}\|_{\infty}\leq 1$ (which is subgaussian by (6)). This is particularly easy. Indeed, for any $y\in[-1,1]$ let

[TABLE]

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

[TABLE]

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 $v:L_{1}(\Omega,\mathbb{P})\to L_{1}([0,1]^{\mathbb{N}}\times\{-1,1\}^{\mathbb{N}})$ 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 $L_{1}(\Omega^{\prime},\mathbb{P}^{\prime})$) for some numerical constant $c$.

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

[TABLE]

More generally, if $\mathcal{B}\subset\mathcal{A}^{\prime}$ is any $\sigma$-subalgebra such that $\mathbb{E}^{\mathcal{B}}\varphi=0$ we have for any $x_{0}\in L_{1}(\Omega^{\prime},\mathcal{B},\mathbb{P}^{\prime};B)$

[TABLE]

[TABLE]

and hence by Jensen

[TABLE]

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

[TABLE]

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

[TABLE]

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 $\mathbb{P}^{\prime}$ replaced by $\mathbb{P}^{\prime}(A)^{-1}\mathbb{P}^{\prime}_{|A}$. The case of a general $A\in\mathcal{B}$ can be proved by a routine approximation argument left to the reader. ∎

We now show that any real valued martingale difference sequence with values in $[-1,1]$ is 1-dominated by $(\varepsilon_{n})$.

[TABLE]

By (13) with $\mathcal{B}=\mathcal{A}_{k-1}$ and $\varphi=d_{k}$ we have

[TABLE]

Now working on the product space $(\Omega,\mathcal{A},\mathbb{P})\times(\Omega^{\prime},\mathcal{A}^{\prime},\mathbb{P}^{\prime})$ with $\mathcal{B}$ equal to $\sigma(\mathcal{A}_{k-2}\cup\varepsilon_{k})$ we find

[TABLE]

Continuing in this way we obtain (16). ∎

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 $2=\int\max\{|1+\varepsilon_{1}|,|1-\varepsilon_{1}|\}d\mathbb{P}\leq\int\max\{|1+z_{1}|,|1-z_{1}|\}d\mu$, which clearly fails.

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

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:

[TABLE]

We claim that for any $N$ and any $z^{0}\in\mathbb{T}^{\mathbb{N}}$ the function $F=\sum\nolimits_{1}^{N}z_{n}^{0}z_{n}\otimes z_{n}$ admits for any $0<\varepsilon\leq 1$ 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

[TABLE]

where we have set

[TABLE]

and where $w(\varepsilon)$ is a function depending only on $0<\varepsilon\leq 1$ (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

[TABLE]

We will view the tensors in $L_{1}(\mathbb{T}^{\mathbb{N}})\otimes L_{1}(\mathbb{T}^{\mathbb{N}})$ as functions of $(z,z^{\prime})\in\mathbb{T}^{\mathbb{N}}\times\mathbb{T}^{\mathbb{N}}$. Note

[TABLE]

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

[TABLE]

Note that (since $\nu_{\varepsilon}\geq 0$ and hence $\|\nu_{\varepsilon}\|_{1}=1$) we have $\|t^{\prime}_{\varepsilon}\|_{\wedge}\leq 2/\varepsilon$. Let $r^{\prime}_{\varepsilon}=\sum\nolimits_{1}^{N}\Re(z_{n}^{0}z_{n}z_{n}^{\prime}))-t^{\prime}_{\varepsilon}$. By the orthogonality in the sum (17) one checks that $\|r^{\prime}_{\varepsilon}\|_{\vee}\leq\varepsilon/2$. This gives us the desired decomposition but, instead of $\sum\nolimits_{1}^{N}{z_{n}^{0}z_{n}z_{n}^{\prime}}$, we are decomposing the sum

[TABLE]

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 $\mathbb{T}$)

[TABLE]

This gives us $\|t_{\varepsilon}\|_{\wedge}\leq 4/\varepsilon$ and $\|r_{\varepsilon}\|_{\vee}\leq\varepsilon$. Moreover we have

[TABLE]

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

We can now complete the proof. Let $(a_{n})$ be a scalar sequence. Let $\Psi=\sum\nolimits_{1}^{N}a_{n}\psi_{n}\otimes\psi_{n}$. Choosing $z_{n}^{0}$ so that $z_{n}^{0}a_{n}=|a_{n}|$ we have

[TABLE]

and hence $\sum|a_{n}|=\langle\Psi,t_{\varepsilon}\rangle+\langle\Psi,r_{\varepsilon}\rangle$ which leads to

[TABLE]

To conclude, we set $C^{\prime}=\sup\nolimits_{n\geq 1}\|\psi_{n}\|_{\infty}$ and we choose, say, $\varepsilon=1/2C^{\prime 2}$. We have then

[TABLE]

∎

Let us say that a bounded set $S$ in $L_{\infty}(T,m)$ is Sidon with constant $C$ if for any finitely supported function $x:S\to\mathbb{C}$ we have $\sum\nolimits_{\varphi\in S}|x(\varphi)|\leq C\|\sum x(\varphi)\varphi\|.$ If $(\varphi_{n})$ is an enumeration of $S$, this is the same as $\sum\nolimits_{n\in\mathbb{N}}|x(n)|\leq C\|\sum\nolimits_{n\in\mathbb{N}}x(n)\varphi_{n}\|.$ 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.

[TABLE]

Then the union $\Lambda_{1}\cup\Lambda_{2}$ is $\otimes^{4}$-Sidon with a constant $C$ depending only on $C_{1},C_{2},C^{\prime}_{1},C^{\prime}_{2},\delta$.

[TABLE]

Then $\|{\mathcal{T}}\|\leq C_{1}+C_{2}$. The operator ${\mathcal{T}}\otimes id_{L_{\infty}(T,m)}$ clearly extends to an bounded operator

[TABLE]

satisfying $\|W\|\leq\|{\mathcal{T}}\|\leq C_{1}+C_{2}$.

We claim that the collection

[TABLE]

is biorthogonal to

[TABLE]

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

[TABLE]

Moreover, if we set $\xi_{n}^{1}=\widetilde{T_{2}}(\varphi_{n}^{1})$ we have

[TABLE]

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. ∎

Acknowledgement. Thanks to Bernard Maurey for useful communications, and to the referee for his/her careful reading.