Sidon sets are proportionally Sidon with small Sidon constants

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definition_2: Hare and Yang's graded weakening of independence: no relation among distinct elements with exponents bounded by n except trivial ones; degree one is quasi-independence, which for sets of positive integers is the dissociation of Problem 774.

lemma_3: In a torsion-free discrete abelian group, each power image E_n = {γ^n : γ ∈ E} of a Sidon set E is Sidon with the same Sidon constant as E.

proposition_2: If the dual group has no nontrivial element of order at most n and the power images E_1, ..., E_n of an identity-free set E are Sidon, one δ_n > 0 gives every finite F ⊆ E an n-degree-independent subset of size at least δ_n|F|.

proposition_4: In a direct sum of cyclic groups of prime orders tending to infinity, a Sidon set has, for each ε > 0, a proportion δ > 0 such that every finite subset contains a subset of at least δ times its size with Sidon constant at most 1 + ε.

theorem_2: Hare and Yang's main theorem: for an identity-free subset of a torsion-free discrete abelian group, being Sidon, being proportionally n-degree independent for each n, and being proportionally Sidon with constant at most 1 + ε for each ε > 0 are equivalent.

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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.

Source identity

The copy read for this card is the arXiv v1 manuscript (stamp "arXiv:1808.03128v1 [math.FA] 9 Aug 2018" on p. 1), the only arXiv version listed on 2026-09-22, 10 physical pages whose printed numbers equal the PDF page numbers (187,189 bytes). Provenance: downloaded from https://arxiv.org/pdf/1808.03128v1 on 2026-09-22. The arXiv record names the published version, Canadian Mathematical Bulletin 62 (2019), 798--809, DOI 10.4153/S0008439518000620 (Cambridge, subscription); that version was not acquired and no version of record was compared, so the labels and pages cited below are v1's: Definition 2 and Remark 1 (p. 3), Theorem 1 (p. 4), Lemma 1 (p. 4), Lemma 2 and Proposition 2 (p. 5), Lemma 3 (p. 7), Theorem 2 (p. 8), Remark 2 and Propositions 3 and 4 (p. 9), Section 4 (pp. 9--10). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1808.03128), every other right reserved.

Read status: claims checked. Definition 2, Remark 1, the proportionality terminology and Theorem 1 (pp. 3--4), Proposition 2, Lemma 3, Theorem 2 and Remark 2 (pp. 5--9) and Proposition 4 (p. 9) were read clause by clause on the arXiv v1 page images; the proof of Lemma 3 was followed, and those of Proposition 2, Theorem 2 and Proposition 4 were read for structure. No proof was independently verified.

Result pages

• Definition 2 (p. 3): $n$-degree and $n$-length independence; degree one is the dissociation of Problem 774.
• Proposition 2 (p. 5): proportional $n$-degree-independent subsets when the power images $E_1,\ldots,E_n$ are Sidon.
• Lemma 3 (p. 7): in a torsion-free group the power images of a Sidon set keep its Sidon constant.
• Theorem 2 (p. 8): the main equivalence in torsion-free groups.
• Proposition 4 (p. 9): the small-constant result in a direct sum of cyclic groups of prime orders tending to infinity.

Terminology

For a subset of a discrete abelian group, written additively here, 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 $i$; in a torsion-free group this means $m_i=0$ unless $\gamma_i$ is the identity (Definition 2 and Remark 1, 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 $\varepsilon>0$, there is $\delta>0$ such that every finite $F\subseteq E$ contains a subset of size at least $\delta|F|$ with Sidon constant at most $1+\varepsilon$.
• Proposition 4 (Section 4). The small-constant conclusion of Theorem 2 also holds for Sidon sets in a direct sum of cyclic groups of prime orders tending to infinity. Proposition 3, credited to Bourgain's methods, states that in $\bigoplus_{i=1}^N\mathbb Z_{p_i}^{\mathbb N}$, with the $p_i$ prime and $p_1$ the least of them, every Sidon set is a finite union of $(p_1-1)$-degree-independent sets; and the paper shows the small-constant conclusion fails in $\mathbb Z_p^{\mathbb N}$ for a fixed prime $p$.

These hypotheses apply to a set $E$ of positive integers in the torsion-free group $\mathbb Z$, whose identity $0$ it avoids. Hence a proportionately dissociated set of positive integers has, for each fixed coefficient bound $n$, a $\delta_n>0$ such that every finite $F\subseteq E$ contains a subset of size at least $\delta_n|F|$ 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 $p$ of degree $N$ with $\widehat p(0)=1$ and $\widehat p(\pm1)\geq1/(1+\varepsilon)$ is composed with each element of an $(N+1)$-degree-independent set and rotated by the phase of the prescribed value there; each such factor is mixed with the constant $1$ in proportion to the modulus of that value, and the factors are multiplied. The degree bound prevents unwanted Fourier collisions, so the product is a probability measure interpolating any prescribed values of modulus at most $1/(1+\varepsilon)$, which bounds the Sidon constant by $1+\varepsilon$.

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.

Bears on.

• #774: Definition 2's degree one is the problem's dissociation for positive integers, and Theorem 2 with Proposition 2 and Lemma 3 gives a proportionately dissociated set of positive integers, for each fixed coefficient bound separately, proportional subsets free of relations with coefficients bounded by it. These are local extraction results; they give no finite partition and settle the problem in neither direction.

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