# On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums

@article{Kolountzakis1998OnTU,
title={On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums},
author={Mihail N. Kolountzakis},
journal={Journal of Number Theory},
year={1998},
volume={76},
pages={147-153}
}
Abstract A set A ⊆{1, …,  N } is of the type B 2 if all sums a + b , with a ⩾ b , a ,  b ∈ A , are distinct. It is well known that the largest such set is of size asymptotic to N 1/2 . For a B 2 set A of this size we show that, under mild assumptions on the size of the modulus m and on the difference N 1/2 −| A | (these quantities should not be too large), the elements of A are uniformly distributed in the residue classes mod  m . Quantitative estimates on how uniform the distribution is are…
9 Citations
Additive correlation and the inverse problem for the large sieve
• B. Hanson
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2018
Abstract Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$. We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S =
New Upper Bounds for Finite Bh Sequences
Abstract Let F h ( N ) be the maximum number of elements that can be selected from the set {1, …,  N } such that all the sums a 1 +…+ a h , a 1 ⩽…⩽ a h are different. We introduce new combinatorial
SIDON SETS IN N
We study finite and infinite Sidon sets in N. The additive energy of two sets is used to obtain new upper bounds for the cardinalities of finite Sidon subsets of some sets as well as to provide short
Sidon sets in Nd
Gaps in dense sidon sets
We prove that if A ⊂ [1, N ] is a Sidon set with N1/2−L elements, then any interval I ⊂ [1, N ] of length cN contains c|A|+EI elements of A, with |EI | ≤ 52N(1+ c1/2N1/8)(1+L + N−1/8), L+ = max{0,
Extremal Sidon sets are Fourier uniform, with applications to partition regularity
• Mathematics
• 2021
Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing
The distribution of dense Sidon subsets of $\mathbb{Z}_m$
Abstract. Let $S \subseteqq \mathbb{Z}_m$ be a Sidon set of cardinality $\mid S \mid = m^{1 \over 2} + O(1)$. It is proved, in particular, that for any interval ${\cal I} = \{a, a + 1, A Complete Annotated Bibliography of Work Related to Sidon Sequences A Sidon sequence is a sequence of integers$a_1 < a_2 < \cdots$with the property that the sums$a_i + a_j(i\le j)\$ are distinct. This work contains a survey of Sidon sequences and their
On the complexity of constructing Golomb Rulers
• Computer Science, Mathematics
Discret. Appl. Math.
• 2009

## References

SHOWING 1-8 OF 8 REFERENCES
Well Distribution of Sidon Sets in Residue Classes
Abstract A set A of non-negative integers is a Sidon set if the sums a + b ( a ,  b ∈ A ,  a ⩽ b ) are distinct. Assume that a ⊆[1,  n ] and that | A |=(1+ o (1)) n 1/2 . Let m ⩾2 be an integer. In
The Density ofBh[g] Sequences and the Minimum of Dense Cosine Sums
A setEof integers is called aBh[g] set if every integer can be written in at mostgdifferent ways as a sum ofhelements ofE. We give an upper bound for the size of aBh[1] subset {n1, …,nk} of {1, …,n}
On a problem of sidon in additive number theory, and on some related problems
• Mathematics
• 1941
To the memory of S. Sidon. Let 0 < a, < a,. .. be an infinite sequence of positive integers. Denote by f(n) the number of solutions of n=a i +a;. About twenty years ago, SIDON 1) raised the question
On Sum Sets of Sidon Sets, 1.
• Mathematics
• 1994
It is proved that there is no Sidon set selected from {1, 2, …,N} whose sum set containsc1N1/2 consecutive integers, but it may containc2N1/3 consecutive integers. Moreover, it is shown that a finite
On sum sets of sidon sets, II
• Mathematics
• 1995
It is proved that there is no Sidon set selected from {1, 2, …,N} whose sum set containsc1N1/2 consecutive integers, but it may containc2N1/3 consecutive integers. Moreover, it is shown that a finite
Theorems in the additive theory of numbers