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… 
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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
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.
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
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
SummaryThis paper extends some earlier results on difference sets andB2 sequences bySinger, Bose, Erdös andTuran, andChowla.
Sequences, Springer-Verlag, New York, 1983
  • Addendum
  • 1996
Sárközy and T . Sós , On sum sets of Sidon sets
  • I , J . Numb . Th .
  • 1994