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Claim. Fix a modulus $m\ge 2$. If $A\subset\{1,\ldots,N\}$ is a Sidon set with $\lvert A\rvert\sim N^{1/2}$, then every residue class modulo $m$ contains $(1/m+o(1))\lvert A\rvert$ elements of $A$ as $N\to\infty$. Lindström [Li98] proves this for $A$ itself by a combinatorial argument, and under the extra assumptions $m=2$ and $\lvert A\rvert\ge N^{1/2}$ bounds the error by $O(N^{3/8})$; these hypotheses are as Kolountzakis reports them (arXiv:math/9808061, p. 1), recorded on the card of Kolountzakis's strengthening, which notes that Lindström states his bound for $m=2$ and $\lvert A\rvert\ge N^{1/2}$ and that Kolountzakis removes both restrictions.

The question of Problem 154 concerns $A+A$, and it follows from the statement for $A$: in a Sidon set distinct unordered pairs $\{a,b\}$ have distinct sums, so the elements of $A+A$ in a residue class $r$ modulo $m$ are in bijection with the unordered pairs whose residues add to $r$, and equidistribution of $A$ among the $m$ classes puts $(1/m+o(1))$ of the pairs in each class. In particular about half the elements of $A+A$ are even and half odd. The site's remark records the same deduction in its own words.

Depends on. No page of this wiki; the result is the paper's.

Acceptance. Refereed: B. Lindström, Well distribution of Sidon sets in residue classes, J. Number Theory 69 (1998), no. 2, 197–200; the issue is dated April 1998, and the page name uses the first day of that month. Reviewed: the site's curator, T. F. Bloom, labels Problem 154 proved at erdosproblems.com on this result and Kolountzakis's strengthening, which is the site's acceptance. Two outside Lean files formalize the argument: the first, posted by Wouter van Doorn to the site's thread on 2026-02-06 and pinned at its commit of 2026-03-02, proves the statement for $A$ (sidon_density_limit) and formalizes, by Harmonic's Aristotle, a write-up of Lindström's proof produced with ChatGPT; the second, first posted on 2026-06-27 and pinned at its commit of 2026-08-22, derives the sumset statement (erdos_154_sumset) from it, and formal-conjectures links both. Neither file has been built or audited in this corpus, so formalized is not listed and the Lean qualification of the site's label is the site's.