Sumsets of Sidon sets

• Autorzy: Imre Z. Ruzsa
• Języki publikacji: EN

Abstrakty

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1996
• Tom: 77
• Numer: 4
• Strony: 353-359

Daty

wydano

1996

otrzymano

1995-11-13

Twórcy

autor

Imre Z. Ruzsa

• Mathematical Institute, Hungarian Academy of Sciences, Budapest, Pf. 127, H-1364 Hungary

Bibliografia

• [1] P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets I, J. Number Theory 47 (1994), 329-347.
• [2] P. Erdős, A. Sárközy and V. T. Sós, On sum sets of Sidon sets II, Israel J. Math. 90 (1995), 221-234.
• [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