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Ruzsa’s Problem on Bi-Sidon Sets
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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
)
.
Title: Ruzsa’s Problem on Bi-Sidon Sets
Description:
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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