# Sumsets of Sidon sets

Acta Arithmetica (1996)

• Volume: 77, Issue: 4, page 353-359
• ISSN: 0065-1036

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## Abstract

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1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence  N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that ${N}^{1/3}$ is possible. In [1] they show that A + A contains gaps longer than clogN, while the maximal gap may be of size O(√N). We improve these bounds. In Section 2, we give an example of A + A containing an interval of length c√N; hence in this question the answer is known up to a constant factor. In Section 3, we construct A such that the maximal gap is $\ll {N}^{1/3}$. In Section 4, we construct A such that the maximal gap of A + A is O(logN) in a subinterval of length cN.

## How to cite

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Imre Z. Ruzsa. "Sumsets of Sidon sets." Acta Arithmetica 77.4 (1996): 353-359. <http://eudml.org/doc/206924>.

@article{ImreZ1996,
abstract = {1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence  N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that $N^\{1/3\}$ is possible. In [1] they show that A + A contains gaps longer than clogN, while the maximal gap may be of size O(√N). We improve these bounds. In Section 2, we give an example of A + A containing an interval of length c√N; hence in this question the answer is known up to a constant factor. In Section 3, we construct A such that the maximal gap is $≪ N^\{1/3\}$. In Section 4, we construct A such that the maximal gap of A + A is O(logN) in a subinterval of length cN.},
author = {Imre Z. Ruzsa},
journal = {Acta Arithmetica},
keywords = {sumsets; Sidon sets; addition of sets; chains; intervals},
language = {eng},
number = {4},
pages = {353-359},
title = {Sumsets of Sidon sets},
url = {http://eudml.org/doc/206924},
volume = {77},
year = {1996},
}

TY - JOUR
AU - Imre Z. Ruzsa
TI - Sumsets of Sidon sets
JO - Acta Arithmetica
PY - 1996
VL - 77
IS - 4
SP - 353
EP - 359
AB - 1. Introduction. A Sidon set is a set A of integers with the property that all the sums a+b, a,b∈ A, a≤b are distinct. A Sidon set A⊂ [1,N] can have as many as (1+o(1))√N elements, hence  N/2 sums. The distribution of these sums is far from arbitrary. Erdős, Sárközy and T. Sós [1,2] established several properties of these sumsets. Among other things, in [2] they prove that A + A cannot contain an interval longer than C√N, and give an example that $N^{1/3}$ is possible. In [1] they show that A + A contains gaps longer than clogN, while the maximal gap may be of size O(√N). We improve these bounds. In Section 2, we give an example of A + A containing an interval of length c√N; hence in this question the answer is known up to a constant factor. In Section 3, we construct A such that the maximal gap is $≪ N^{1/3}$. In Section 4, we construct A such that the maximal gap of A + A is O(logN) in a subinterval of length cN.
LA - eng
KW - sumsets; Sidon sets; addition of sets; chains; intervals
UR - http://eudml.org/doc/206924
ER -

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