# Generalized sidon sets

@article{Cilleruelo2009GeneralizedSS, title={Generalized sidon sets}, author={Javier Cilleruelo and Imre Z. Ruzsa and Carlos Vinuesa}, journal={Advances in Mathematics}, year={2009}, volume={225}, pages={2786-2807} }

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.

#### 22 Citations

On Additive Representative Functions

- Mathematics, Computer Science
- The Mathematics of Paul Erdős I
- 2013

This paper gives a short survey of additive representation functions on their regularity properties and value distribution and proves a couple of new results and presents many related unsolved problems. Expand

Sidon sets in Nd

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2010

The additive energy of two sets is used to obtain new upper bounds for the cardinalities of finite Sidon subsets of some sets as well as to provide short proofs of already known results. Expand

Improved bounds on the supremum of autoconvolutions

- Mathematics
- 2009

We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we… Expand

Popular differences and generalized Sidon sets

- Mathematics
- 2017

Abstract For a subset A ⊆ [ N ] , we define the representation function r A − A ( d ) : = # { ( a , a ′ ) ∈ A × A : d = a − a ′ } and define M D ( A ) : = max 1 ≤ d D r A − A ( d ) for D > 1 . We… Expand

On an application of higher energies to Sidon sets

- Mathematics
- 2021

Sidon sets is a classical object of Combinatorial Number theory, which was introduced by S. Sidon in [24]. A subset S of an abelian group G is a Sidon set iff all its non–zero differences are… Expand

Threshold functions and Poisson convergence for systems of equations in random sets

- Mathematics
- 2018

We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets,… Expand

On symmetric intersecting families

- Mathematics, Computer Science
- Eur. J. Comb.
- 2020

It is proved that a symmetric intersecting family of k-element subsets of 1,2, n of sets has size at most where $c > 0$ is a universal constant. Expand

SYMMETRY AND COLORINGS: SOME RESULTS AND OPEN PROBLEMS, II

- Mathematics
- 2011

We survey some principal results and open problems related to colorings of geometric and algebraic objects endowed with symmetries. We concentrate the exposition on the maximal symmetry numbers… Expand

On sidon sequences of farey sequences, square roots and reciprocals

- Mathematics
- 2020

Abstract In this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that… Expand

Generalized difference sets and autocorrelation integrals

- Mathematics
- 2020

In 2010, Cilleruelo, Ruzsa, and Vinuesa established a surprising connection between the maximum possible size of a generalized Sidon set in the first $N$ natural numbers and the optimal constant in… Expand

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Sidon sets in Nd

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We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we… Expand