Arithmetic characterizations of Sidon sets

../

proposition_p88: The conditions of Theorem 1 are also equivalent to an exponentially small measure for the set where every character of A has real part above rho, and to the existence of exponentially many points of G separated by alpha in the sup over A.

theorem_1: For a subset of a discrete abelian group not containing 0, being Sidon is equivalent to each of three uniform bounds, of the form 2^(theta|A|) or 3^(theta|A|) with theta < 1, on the signed representation counts of its finite subsets A.

theorem_2: A subset of a discrete abelian group is Sidon if and only if, for some integer k, every finite subset A contains a quasi-independent subset of size at least |A|/k; for infinite sets of positive integers this is proportionate dissociation as in Problem 774.

---

Gilles Pisier, “Arithmetic characterizations of Sidon sets,” Bulletin of the American Mathematical Society (New Series) 8 (1983), no. 1, 87–89; DOI 10.1090/S0273-0979-1983-15092-9.

Copy read. The copy read for this card is the published article, which prints "© 1983 American Mathematical Society 0273-0979/82/0000-1035/$01.75" in its first-page footer, every other right reserved.

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, (v) and (vi). 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, and that (v) $\Rightarrow$ (i) answers Problem 8.3 of its reference [4]. It also notes that the equivalence of (iii) and (iv) follows easily from $\sum_\gamma R(\gamma,A)=3^{|A|}$.

Result pages: Theorem 1 and the Proposition.

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 an infinite $\Lambda\subset\mathbb N$ is proportionately dissociated in the sense of E0774 if and only if it is Sidon; the translation is this card's, and is spelled out on the result page Theorem 2, which also records the Definition of quasi-independent and Rider sets (p. 88). 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). Restricted to sets of positive integers, under the 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.
• Theorem 2 permits either side of a proposed construction to be checked in a different language. Proportional dissociation can be established directly by extracting a large relation-free subset, or indirectly by proving that the resulting integer set is Sidon.
• Theorem 1 supplies finite obstructions to Sidonicity. To prove that a candidate is proportionately dissociated, it is enough in principle to obtain a uniform exponential saving in one of the relation counts. To disprove proportional dissociation, one should seek finite blocks for which these counts approach the unrestricted exponents $2^{|A|}$ or $3^{|A|}$.
• For the desired counterexample, the two requirements are therefore sharply separated: retain one of the uniform Sidon/extraction bounds, while forcing the quasi-independent chromatic number of finite subconfigurations to be unbounded. Pisier’s equivalence validates this target but does not construct it.
• 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 implication (i) $\Rightarrow$ (ii) 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 implication from Sidon to (vii) in 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 $\widehat{\mathbb T}=\mathbb Z$.

Bears on. E0774 and E0963. Theorem 2 makes the hypothesis of E0774, for an infinite set of positive integers, equivalent to its being Sidon, and E0774 is Pisier’s concluding question (p. 89) restricted to sets of positive integers, where quasi-independent means dissociated; the paper does not answer it. For E0963 the paper gives context only: by Theorem 2, the finite subsets of one Sidon set contain quasi-independent (in E0963’s terms, dissociated) subsets of linear size, while E0963 asks how large a dissociated subset every $n$-element set of reals must contain.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.