# Sumsets of Sidon sets

Acta Arithmetica (1996)

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

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topImre 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 -

## References

top- [1] P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets I, J. Number Theory 47 (1994), 329-347. Zbl0811.11014
- [2] P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets II, Israel J. Math. 90 (1995), 221-234. Zbl0841.11006
- [3] H. Halberstam and K. F. Roth, Sequences, Clarendon, 1966.
- [4] I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith. 65 (1993), 259-282 Zbl1042.11525

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