The Sidon Decomposition Problem in Abelian Groups of Bounded Torsion

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Mark Lewko, “The Sidon Decomposition Problem in Abelian Groups of Bounded Torsion,” arXiv:2606.06669v1 (2026). The arXiv record (https://arxiv.org/abs/2606.06669, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Main result

Theorem 1.2 (p. 2) reads: "Let $G$ be a compact abelian group whose dual group $\Gamma=\widehat G$ has bounded torsion. Then a set $\Lambda\subseteq\Gamma\setminus\{0\}$ is Sidon if and only if $\Lambda$ is a finite union of quasi-independent sets." Bounded torsion means that some integer $N$ satisfies $N\gamma=0$ for every $\gamma\in\Gamma$ (p. 2). A set $Q\subseteq\Gamma\setminus\{0\}$ is quasi-independent when the only finitely supported $\epsilon\in\{-1,0,1\}^Q$ with $\sum_{\gamma\in Q}\epsilon_\gamma\gamma=0$ is $\epsilon=0$ (p. 1).

The quantitative core is the following finite prime-power statement. Let $q=p^s$ and $A\subseteq(\mathbb Z/q\mathbb Z)^n$. If there is a $\delta>0$ such that every $B\subseteq A$ contains a quasi-independent $Q\subseteq B$ with $|Q|\geq\delta|B|$, then

$$A=\bigsqcup_{i=1}^k A_i, \qquad k\leq\left\lceil\frac{s\log_2 p}{\delta}\right\rceil,$$

with every $A_i$ quasi-independent. This is Theorem 1.3 (p. 2), proved in Section 5 (pp. 7--8). The bound is uniform in $n$ and $|A|$. Bourgain's projection theorem (Theorem 6.1, p. 8) reduces a Sidon set in a bounded-torsion group to finitely many Sidon sets in groups of exponent dividing a fixed prime power, Pisier's arithmetic characterization supplies the proportional hypothesis for finite subsets of a Sidon set (Theorem 1.1, p. 1), and a compactness argument turns the uniform finite coloring into an infinite decomposition (Section 6, pp. 9--10).

Mechanisms potentially useful for E0774

For subsets of $\mathbb N\subset\mathbb Z$, dissociation in E0774 is the same signed-relation condition as quasi-independence: canceling the intersection of two unequal finite subsets converts an equality of subset sums into a nonzero $\{-1,0,1\}$ relation, and conversely.

Lewko's proof packages those relations in a way that suggests a route to a finite form of E0774:

1. For each $T\subseteq A$, collect the signed zero-relations supported on $T$. Proportional quasi-independence gives at least $2^{\delta|T|}$ distinct subset sums and hence growth of the subgroup generated by $T$ (Section 3, p. 3).
2. In exponent $p^s$, compare that growth with the finite coefficient space $(\mathbb Z/p^s\mathbb Z)^T$. Kernel counting and reduction modulo $p$ show that the reduced signed relations supported on $T$ span a space of dimension at most

$$\left(1-\frac{\delta}{s\log_2 p}\right)|T|.$$

This is Corollary 3.3 (p. 4). 3. A support-partition theorem then turns any uniform local dimension gap into a coloring: if $X$ is finite, $k\geq2$, and a collection $\mathcal C\subseteq\mathbb F^X\setminus\{0\}$ over a field $\mathbb F$ satisfies $\dim\operatorname{span}{c\in\mathcal C:\operatorname{supp}c\subseteq Y} \leq (k-1)|Y|/k$ for every $Y\subseteq X$, then $X$ can be partitioned into $k$ classes, none containing the support of a member of $\mathcal C$. This is Theorem 4.3 (p. 6), derived from Rado--Horn. 4. Because every nonzero signed coefficient $\pm1$ stays nonzero modulo $p$, the support-avoiding coloring rules out the original signed relations and makes every color class quasi-independent (Section 5, pp. 7--8).

The most portable component for E0774 is therefore Theorem 4.3: it separates the coloring step from the arithmetic step. A transfer to $\mathbb Z$ would follow from a suitable field-valued representation of the integer signed relations together with a local span-dimension gap bounded away from $|T|$. The finite-to-infinite compactness step is also directly reusable once the number of colors is uniform.

Why the theorem does not settle E0774

The paper explicitly says that the analogous problem for $\Gamma=\mathbb Z$ is not addressed (Section 1, p. 2). The exclusion is structural, not terminological: $\mathbb Z$ has no bounded exponent, so it has neither the finite coefficient space $(\mathbb Z/p^s\mathbb Z)^T$ nor a fixed prime-power coordinate factor to which Bourgain's projection theorem can reduce the problem.

In particular, the decisive count $|\ker\Phi_T|\,|\langle T\rangle|=q^{|T|}$ takes place between finite groups. For a nonempty $T\subset\mathbb Z\setminus\{0\}$, the generated subgroup $\langle T\rangle$ is infinite, so distinct subset sums do not yield the same finite-cardinality bound on the relation kernel. Reducing the integers modulo a prime does not repair this uniformly: it introduces modular zero-relations that need not be integer zero-relations, while no single modulus is supplied by the E0774 hypothesis. Thus the Rado--Horn partition mechanism remains relevant, but the bounded-torsion argument that establishes its dimension hypothesis is exactly the missing transfer step for E0774.

Bears on. #774