Problem 154

../

claims/: The 2 claim pages of Problem 154, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $A\subset \{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert\sim N^{1/2}$. Must $A+A$ be well-distributed over all small moduli? In particular, must about half the elements of $A+A$ be even and half odd?

Formulation. The wording does not say whether "all small moduli" means each fixed modulus or moduli that grow with $N$. The sources read it as each fixed modulus $m\ge2$: Kolountzakis (arXiv:math/9808061, p. 1) says that Lindström's theorem for a constant modulus answers the question of Erdős, Sárközy and Sós, and the site's remark accepts that answer. In that reading, for every residue $r$ the proportion of the elements of $A+A$ congruent to $r$ modulo $m$ tends to $1/m$ as $N\to\infty$, and the standing below answers it. Kolountzakis's bounds also allow moduli that grow with $N$, but only in ranges that depend on $N^{1/2}-\lvert A\rvert$, as his claim page states; for a set known only to satisfy $\lvert A\rvert\sim N^{1/2}$ they give no range of moduli up to a fixed power of $N$.

Status. PROVED (LEAN), the site's label. The standing rests on two accepted claim pages: Lindström 1998, the refereed equidistribution of $A$ itself in residue classes, from which the statement for $A+A$ follows by the Sidon property, and its refereed quantitative strengthening Kolountzakis 1999. The label's Lean qualification dates from Wouter van Doorn's formalization of the statement for $A$, posted to the site's thread on 2026-02-06; a formal derivation of the sumset statement from it followed on 2026-06-27. Both are linked from Lindström's page, and neither has been built or audited in this corpus.

Source. erdosproblems.com/154, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #154, https://www.erdosproblems.com/154.

References.

• [Ko99] Kolountzakis, Mihail N., On the uniform distribution in residue classes of dense sets of integers with distinct sums. J. Number Theory (1999), 147-153.
• [Li98] Lindström, Bernt, Well distribution of Sidon sets in residue classes. J. Number Theory (1998), 197-200.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• kolountzakis_1999_uniform_distribution_residue_classes_dense_sets
• kolountzakis_1999_uniform_distribution_residue_classes_dense_sets / theorem_2