Ruzsa’s Problem on Bi-Sidon Sets

Pach, János and Zakharov, Dmitrii (2025) Ruzsa’s Problem on Bi-Sidon Sets. COMBINATORICA, 45 (2). ISSN 0209-9683

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Abstract

A subset S of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of S are distinct. Imre Ruzsa asked the following question: What is the maximum number f ( N ) such that every set S of N real numbers contains a bi-Sidon subset of size at least f ( N )? He proved that f(N)\geqslant cN^{\frac{1}{3}} f ( N ) ⩾ c N 1 3 , for a constant c>0 c > 0 . In this note, we improve this bound to N^{\frac{1}{3}+\frac{7}{78}+o(1)} N 1 3 + 7 78 + o ( 1 ) .

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