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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_151_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$f(N)\geqslant cN^{\frac{1}{3}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>⩾</mo> <mi>c</mi> <msup> <mi>N</mi> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </msup> </mrow> </math></EquationSource> </InlineEquation>, for a constant <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_151_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this note, we improve this bound to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_151_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$N^{\frac{1}{3}+\frac{7}{78}+o(1)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>N</mi> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mn>7</mn> <mn>78</mn> </mfrac> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </math></EquationSource> </InlineEquation>.

Ruzsa’s Problem on Bi-Sidon Sets

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2025年4月8日

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János Pach,
Dmitrii Zakharov

摘要

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 $f(N)\geqslant cN^{\frac{1}{3}}$ $f (N) ⩾ c N^{\frac{1}{3}}$ , for a constant $c>0$ $c > 0$ . In this note, we improve this bound to $N^{\frac{1}{3}+\frac{7}{78}+o(1)}$ $N^{\frac{1}{3} + \frac{7}{78} + o (1)}$ .