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8 Acta Arithmetica

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On the additive completion of primes

100%

Ruzsa I.

Acta Arithmetica

1998

tom 86

nr 3

269-275

Sumsets of Sidon sets

100%

Ruzsa I.

Acta Arithmetica

1996

tom 77

nr 4

353-359

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.

Arithmetic progressions in sumsets

100%

Ruzsa I.

Acta Arithmetica

1991-1992

tom 60

nr 2

191-202

1. Introduction. Let A,B ⊂ [1,N] be sets of integers, |A|=|B|=cN. Bourgain [2] proved that A+B always contains an arithmetic progression of length $exp(logN)^{1/3-ε}$. Our aim is to show that this is not very far from the best possible. Theorem 1. Let ε be a positive number. For every prime p > p₀(ε) there is a symmetric set A of residues mod p such that |A| > (1/2-ε)p and A + A contains no arithmetic progression of length (1.1)} $exp(logp)^{2/3+ε}$. A set of residues can be used to get a set of integers in an obvious way. Observe that the 1/2 in the theorem is optimal: if |A|>p/2, then A+A contains every residue. Acknowledgement. I profited much from discussions with E. Szemerédi; he directed my attention to this problem and to Bourgain's paper.

Negative values of cosine sums

100%

Ruzsa I.

Acta Arithmetica

2004

tom 111

nr 2

179-186

Solving a linear equation in a set of integers II

100%

Ruzsa I.

nr 4

385-397

Solving a linear equation in a set of integers I

100%

Ruzsa I.

Acta Arithmetica

1993

tom 65

nr 3

259-282

A set of squares without arithmetic progressions

63%

Gyarmati K. , Ruzsa I.

Acta Arithmetica

2012

tom 155

nr 1

109-115

Prime values of reducible polynomials, II

45%

Chen Y.-G. , Kun G. , Pete G. , Ruzsa I. , Timar A.

tom 104

nr 2

117-127