New Upper Bounds for Finite Bh Sequences

@article{Cilleruelo2001NewUB,
  title={New Upper Bounds for Finite Bh Sequences},
  author={Javier Cilleruelo},
  journal={Advances in Mathematics},
  year={2001},
  volume={159},
  pages={1-17}
}
  • J. Cilleruelo
  • Published 15 April 2001
  • Mathematics
  • Advances in Mathematics
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 and analytic ideas to prove new upper bounds for F h ( N ). In particular we prove F 3 (N)⩽ 4 1+16/( π +2) 4 N 1/3 +o(N 1/3 ), F 4 (N)⩽ 8 1+16/( π +2) 4 N 1/4 +o(N 1/4 ). Besides, our techniques have an independent interest for further research in additive number theory. 
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16 Citations

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References

SHOWING 1-10 OF 11 REFERENCES
Upper and Lower Bounds for Finite Bh[g] Sequences
Abstract We give a non-trivial upper bound for Fh(g,N), the size of a Bh[g] subset of {1,…,N}, when g>1. In particular, we prove F2(g,N)⩽1.864(gN)1/2+1, and F h (g,N)⩽ 1 (1+cos h (π/h)) 1/h (hh!gN)…
On Finite Sidon Sequences
Abstract A set A of integers is called a Bh-sequence if all sums a1 + · · · + ah, where ai ∈ A, are distinct up to rearrangement of the summands. Let Fh(n) (resp. ƒh(n)) denote the size of a largest…
On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums
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…
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…
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,…
On the size of finite Sidon sequences
Let h > 2 be -an integer. A set of positive integers B is called a Bh-sequence, or a Sidon sequence of order h, if all sums aI + a2 + * + ah, where ai E B (i = 1, 2, ..., h), are distinct up to…
B h sequences
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…
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}…
A Remark on B2k-Sequences
  • M. Helm
  • Mathematics, Computer Science
  • 1994
TLDR
It is proved that lim infn → ∞A(n)/n1/2k log n1/(3k − 1) is correct and B2k-sequences can be sequenced using LaSalle's inequality.
Theorems in the additive theory of numbers
SummaryThis paper extends some earlier results on difference sets andB2 sequences bySinger, Bose, Erdös andTuran, andChowla.
...
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