Sidon sets are proportionally Sidon with small Sidon constants

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Kathryn E. Hare and Robert (Xu) Yang, “Sidon sets are proportionally Sidon with small Sidon constants,” Canadian Mathematical Bulletin 62 (2019), 798--809; arXiv:1808.03128.

Terminology

For a subset of a torsion-free discrete abelian group, the paper calls a set $n$-degree independent if every relation

$$\sum_i m_i\gamma_i=0, \qquad |m_i|\leq n,$$

on distinct elements has $m_i\gamma_i=0$ for every $i$, so $m_i=0$ unless $\gamma_i$ is the identity (Definition 2, p. 3). Degree one is called quasi-independence. For sets of positive integers this is exactly the property called dissociation in Problem 774: cancelling the intersection of two equal subset sums produces a nonzero relation with coefficients in $\{-1,0,1\}$, and conversely. The paper reserves dissociate for the stronger degree-two property (Definition 2, Section 2).

Results relevant to E0774

Theorem 1(a) recalls Pisier's equivalence: a set not containing the identity is Sidon if and only if every finite subset contains a quasi-independent subset of at least a fixed positive proportion. Thus the hypothesis in E0774 is precisely Sidonicity for subsets of the positive integers. Section 2 (p. 3) explicitly lists as open whether every Sidon set is a finite union of quasi-independent sets.

The paper strengthens the local side of this equivalence.

• Proposition 2 (Section 3). Fix $n\geq1$. Suppose the ambient group has no nontrivial element of order at most $n$, $E$ does not contain the identity, and every power image $E_k=\{\gamma^k:\gamma\in E\}$, $1\leq k\leq n$, is Sidon. Then there is $\delta_n>0$ such that every finite $F\subseteq E$ contains an $n$-degree-independent $H$ with $|H|\geq\delta_n|F|$.
• Lemma 3 (Section 3). In a torsion-free group, if $E$ is Sidon, then every $E_k$ is Sidon with the same Sidon constant as $E$.
• Theorem 2 (Section 3). For a torsion-free group and $E$ not containing the identity, the following are equivalent: $E$ is Sidon; for every fixed $n$, $E$ is proportionally $n$-degree independent; and, for every $\varepsilon>0$, every finite subset of $E$ contains a linearly large subset with Sidon constant at most $1+\varepsilon$.

These hypotheses apply to $E\subseteq\mathbb N\subseteq\mathbb Z$. Hence a proportionately dissociated integer set has, for each fixed coefficient bound $n$, linearly large subsets avoiding every relation with coefficients bounded by $n$.

Methods

Lemma 1 turns Sidonicity into a subgaussian exponential-moment estimate. Lemma 2 applies it simultaneously to the first $n$ power images. In the proof of Proposition 2, the authors randomly thin a finite set, bound the number of bounded-coefficient relations by a Riesz-product integral, and delete a maximal relation set. An entropy comparison ensures that a positive fraction survives. This provides a quantitative way to certify large good subsets without enumerating all subsets.

For Theorem 2, a positive trigonometric polynomial approximating a prescribed phase is multiplied over an $(N+1)$-degree-independent set. The degree bound prevents unwanted Fourier collisions, producing an interpolating measure of norm at most $1+\varepsilon$.

Limit for the open problem

All conclusions are local. The extracted subset may depend on the finite set, and $\delta_n$ may deteriorate with $n$. Repeated extraction yields a number of pieces that grows with the size of a finite set; it does not produce a uniform coloring of the infinite signed-relation hypergraph. Bourgain's finite-union result quoted in Section 2 concerns bounded relation length, not arbitrary support, so it also does not settle E0774.

The paper is therefore useful both as an extraction toolkit and as a sharp description of the missing step: even simultaneous proportional avoidance of every fixed coefficient bound has not been converted into a finite global partition by quasi-independent sets.