Gaps in dense sidon sets
@inproceedings{Cilleruelo2000GapsID, title={Gaps in dense sidon sets}, author={Javier Cilleruelo}, year={2000} }
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, L}. In particular, if |A| = N + O(N), and g(A) is the maximum gap in A, we deduce that g(A) ? N. Also we prove that, under this condition, the exponent 3/4 is sharp.
12 Citations
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References
SHOWING 1-10 OF 11 REFERENCES
ON A PROBLEM OF SIDON IN ADDITIVE NUMBER THEORY, AND ON SOME RELATED PROBLEMS
- Mathematics
- 2002
Let a,<&<... be a sequence of positive integers, and suppose that the suma czi+lzi (where i ,<j) are all different. Such sequences, called B, sequences by Sidont, occur in the theory of Fourier…
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 the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums
- Mathematics
- 1998
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…
B h sequences
- Mathematics
- 1996
Let A be a set (finite or infinite) of natural numbers, and let a i denote a generic element of A. We say that A is a B h sequence if the sums of the form a 1 + … + a h (a 1 ≤ … ≤ a h ) are all…
Solving a linear equation in a set of integers I
- Mathematics
- 1993
(1.1) a1x1 + . . .+ akxk = b with x1, . . . , xk in a prescribed set of integers. We saw that the vanishing of the constant term b and the sum of coefficients s = a1 + . . . + ak had a strong effect…
Analytic Number Theory : Proceedings of a Conference In Honor of Heini Halberstam Volume 1
- Mathematics
- 1996
An estimate for Heilbronn's exponential sum, D.R. Heath-Brown On B3 sequences, M. Helm On an elementary inequality in the theory of Diophantine approximation, C. Hooley The integer points close to a…
Theorems in the additive theory of numbers
- Business
- 1962
SummaryThis paper extends some earlier results on difference sets andB2 sequences bySinger, Bose, Erdös andTuran, andChowla.
Chowla, Theorems in the additive theory of numbers, Comment.math.helvet
- 1962