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Claim. Let $A\subset\{1,\ldots,N\}$ be a Sidon set with $\lvert A\rvert=k\ge N^{1/2}-l(N)$, where $l=o(N^{1/2})$, and let $m=o(N^{1/2})$. Writing $a(x)$ for the number of elements of $A$ congruent to $x$ modulo $m$, Theorem 2 of [[../library/additive_bases/kolountzakis_1999_uniform_distribution_residue_classes_dense_sets/_index|the paper]] bounds the $\ell^2$ discrepancy $\lVert a(x)-k/m\rVert_2$ by $CN^{3/8}/m^{1/4}$ when $l\le N^{1/4}m^{1/2}$ and by $CN^{1/4}l^{1/2}/m^{1/2}$ otherwise, and its remarks deduce $a(x)=k/m+o(k/m)$ uniformly in $x$ for $m=o(N^{1/6})$ in the first range and $m=o(N^{1/2}/l)$ in the second. For a fixed modulus and $\lvert A\rvert\sim N^{1/2}$ this recovers Lindström's equidistribution of $A$, and for constant $m$ and $l\le CN^{1/4}$ it bounds the $\ell^2$ discrepancy by $C_mN^{3/8}$, the error Lindström obtained only for $m=2$ and $\lvert A\rvert\ge N^{1/2}$; the statement for $A+A$ follows by the Sidon property as the page of [[problems/additive_bases/E0154/claims/1998_04_01_lindstrom|Lindström's claim]] explains. The method is analytic: the input is the author's earlier estimate for nonnegative cosine polynomials with distinct integer frequencies.

Depends on. [[problems/additive_bases/E0154/claims/1998_04_01_lindstrom|Lindström's page]] for the deduction of the sumset statement from the equidistribution of $A$; the equidistribution itself is the paper's.

Acceptance. Refereed: M. N. Kolountzakis, On the uniform distribution in residue classes of dense sets of integers with distinct sums, J. Number Theory 76 (1999), no. 1, 147–153; the page name uses the date of the first version of the preprint, arXiv:math/9808061, 1998-08-14. Reviewed: the site's curator, T. F. Bloom, records the problem as proved at erdosproblems.com on Lindström's theorem and this strengthening, which is the site's acceptance. No Lean formalization of this quantitative statement is recorded; the formalizations linked from Lindström's page cover the qualitative statement.