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1
Content available remote

Solving a linear equation in a set of integers I

100%
Ruzsa I.
Acta Arithmetica
|
1993
|
tom 65
|
nr 3
259-282
2
Content available remote

Arithmetic progressions in sumsets

100%
Ruzsa I.
Acta Arithmetica
|
1991-1992
|
tom 60
|
nr 2
191-202
EN
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.
3
Content available remote

Solving a linear equation in a set of integers II

100%
Ruzsa I.
Acta Arithmetica
|
1995
|
tom 72
|
nr 4
385-397
4
Content available remote

On the additive completion of primes

100%
Ruzsa I.
Acta Arithmetica
|
1998
|
tom 86
|
nr 3
269-275
5
Content available remote

Negative values of cosine sums

100%
Ruzsa I.
Acta Arithmetica
|
2004
|
tom 111
|
nr 2
179-186
6
Content available remote

Sumsets of Sidon sets

100%
Ruzsa I.
Acta Arithmetica
|
1996
|
tom 77
|
nr 4
353-359
EN
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.
7
Content available remote

A set of squares without arithmetic progressions

64%
Gyarmati K., Ruzsa I.
Acta Arithmetica
|
2012
|
tom 155
|
nr 1
109-115
8
Content available remote

Difference sets and the primes

64%
Ruzsa I., Sanders T.
Acta Arithmetica
|
2008
|
tom 131
|
nr 3
281-301
9
Content available remote

Prime values of reducible polynomials, II

45%
Chen Y.-G., Kun G., Pete G., Ruzsa I., Timar A.
Acta Arithmetica
|
2002
|
tom 104
|
nr 2
117-127
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