Arithmetic characterizations of Sidon sets

Library card.

---

Gilles Pisier, “Arithmetic characterizations of Sidon sets,” Bulletin of the American Mathematical Society (New Series) 8 (1983), no. 1, 87–89.

Scope and notation

Let $G$ be a compact abelian group and let $\widehat G$ be its discrete dual. For $\Lambda\subset\widehat G$, Pisier writes $R_\Lambda$ for the finitely supported relations

$$\sum_{\lambda\in\Lambda}\epsilon_\lambda\lambda=0, \qquad \epsilon_\lambda\in\{-1,0,1\}.$$

The larger set $I_\Lambda$ consists of all finitely supported $\{-1,0,1\}$-families, whether or not they give a zero relation. For $\gamma\in\widehat G$, $R(\gamma,\Lambda)$ counts the representations $\gamma=\sum\epsilon_\lambda\lambda$ coming from $I_\Lambda$, and $R_s(\gamma,\Lambda)$ counts those with $\sum|\epsilon_\lambda|=s$.

A set $\Lambda$ is quasi-independent when $R(0,\Lambda)=1$: its only $\{-1,0,1\}$ relation is the zero relation. It is a Rider set when, for some $\delta>0$, $\sum_{s\geq 0}\delta^sR_s(0,\Lambda)<\infty$. A Sidon set is a set $\Lambda$ for which there is $K$ such that every trigonometric polynomial $f$ with Fourier support in $\Lambda$ satisfies

$$\sum_\gamma |\widehat f(\gamma)|\leq K\|f\|_{C(G)}.$$

The least such $K$ is denoted $S(\Lambda)$. These definitions are on pp. 87–88 (the article's first two pages).

For $\Lambda\subset\mathbb N\subset\mathbb Z=\widehat{\mathbb T}$, quasi-independence is exactly dissociation in the sense of E0774. Indeed, a nonzero signed relation partitions its support into two distinct finite subsets with equal sums. Conversely, equality of two distinct subset sums, after cancelling their intersection, gives a nonzero signed relation.

Main arithmetic characterizations

Theorem 1 (p. 88; the article's second page). Suppose $0\notin\Lambda\subset\widehat G$. The following are equivalent:

• (i) $\Lambda$ is Sidon.
• (ii) For some $\theta<1$, every finite $A\subset\Lambda$ satisfies

$$\sum_{s\geq0}2^{-s}R_s(0,A)\leq 2^{\theta|A|}.$$

• (iii) For some $\theta<1$, every finite $A\subset\Lambda$ satisfies

$$\sup_{\gamma\in\widehat G}R(\gamma,A)\leq3^{\theta|A|}.$$

• (iv) For some $\theta<1$, every finite $A\subset\Lambda$ satisfies

$$\left(\sum_{\gamma\in\widehat G}R(\gamma,A)^2\right)^{1/2} \leq3^{\theta|A|}.$$

The accompanying proposition (p. 88; the article's second page) says these conditions are also equivalent to two analytic/metric formulations. First, there are $\alpha>0$ and $\rho<1$ such that, for every finite $A\subset\Lambda$,

$$m\{t\in G:\inf_{\lambda\in A}\operatorname{Re}\lambda(t)>\rho\} \leq2^{-\alpha|A|}.$$

Second, there is $\alpha>0$ such that for every such $A$ there are $N\geq2^{\alpha|A|}$ points $t_1,\dots,t_N\in G$ with

$$\sup_{\lambda\in A}|\lambda(t_i)-\lambda(t_j)|\geq\alpha \quad(i\ne j).$$

The paper says the equivalence of these last two formulations is formal. It also notes that the equivalence of the two representation-count bounds follows easily from $\sum_\gamma R(\gamma,A)=3^{|A|}$.

Theorem 2 (p. 89; the article's third page). “A subset $\Lambda$ of $\widehat G$ is a Sidon set iff (vii) there is an integer $k$ such that any finite subset $A$ of $\Lambda$ contains a quasi-independent subset $B\subset A$ with $|B|\geq|A|/k$.”

Thus, for $A\subset\mathbb N$, “proportionately dissociated” is equivalent to “Sidon.” Pisier says that Theorem 2, in some sense, reduces the outstanding finite-union problem to “a purely combinatorial question: Is every set satisfying (vii) a finite union of quasi-independent sets?” (p. 89). Under the integer translation above, that is precisely E0774.

Proof ideas useful for E0774

• A union of $k$ quasi-independent sets has the proportional extraction property immediately: one color class contains at least $|A|/k$ elements of every finite $A$. E0774 asks for the converse.
• The identity displayed as (1) on p. 88 expands $\prod_{\lambda\in A}[1+\delta(\lambda+\bar\lambda)]$ in terms of the weighted counts $R_s(\gamma,A)$. Pisier says the Sidon-to-relation-count implication uses this identity at $\delta=1/2$ together with integrability properties of $\sum_{\lambda\in A}\operatorname{Re}\lambda$.

Exact limits of this source

• This three-page article is an announcement. It sends the details of Theorem 1, the proposition, and the difficult direction of Theorem 2 to Pisier’s reference [5], then listed as forthcoming. It does not contain those proofs.
• The converse direction in Theorem 2 is attributed to Theorem 2.3 of Pisier’s 1981 paper, using the fact that every quasi-independent set has Sidon constant bounded by an absolute constant.
• The paper states that every Rider set is a finite union of quasi-independent sets, again referring to [5] rather than proving it here.
• No explicit dependence of $k$, $\theta$, $\alpha$, or $\rho$ on the Sidon constant is given. The equivalences are qualitative and uniform over finite subsets.
• Pisier does not answer the finite-union question and provides no integer counterexample, block construction, or encoding scheme. The positive results it mentions, for $G=\mathbb Z(p)^{\mathbb N}$ with $p$ prime (its reference [3]) or a product of distinct primes (Bourgain, private communication), do not cover $\widehat{\mathbb T}=\mathbb Z$.