• Corpus ID: 44313019


  title={SIDON SETS IN N},
  author={Javier Cilleruelo},
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 proofs of already known results. We also disprove a conjecture of Lindstrom on the largest Sidon set in [1, N ]× [1, N ] and relate it to a known conjecture of Vinogradov concerning the size of the smallest quadratic residue modulo a prime p. For infinite Sidon sets A ⊂ N, we prove that lim infn… 
Sum of elements in finite Sidon sets II
. A set S ⊂ { 1 , 2 , ..., n } is called a Sidon set if all the sums a + b ( a, b ∈ S ) are different. Let S n be the largest cardinality of the Sidon sets in { 1 , 2 , ..., n } . In a former article,
An upper bound on the size of Sidon sets
It is shown that the maximum size of a Sidon set of {1, 2, . . . , n} is at most √n+ 0.998n for sufficiently large n.
A note on Sidon-Ramsey numbers
Given a positive integer k, the Sidon-Ramsey number SR(k) is defined as the minimum n such that in every partition of [1, n] into k parts there is a part containing two pairs of numbers with the same
We study sets of integers A with |A− A| close to |A| and prove that |A| < √ n + √ |A|2 − |A−A| for any set A ⊂ {1, . . . , n}. For infinite sequences of positive integers A = (an) we define An = {a1,
On Sidon sets in a random set of vectors
For positive integers $d$ and $n$, let $[n]^d$ be the set of all vectors $(a_1,a_2,\dots, a_d)$, where $a_i$ is an integer with $0\leq a_i\leq n-1$. A subset $S$ of $[n]^d$ is called a \emph{Sidon
Generalization of a theorem of Erdős and Rényi on Sidon sequences
Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(ϵ) and a sequence of integers A such that the number of ordered representations of any number as a sum
Sets of integers avoiding congruent subsets
  • R. Tesoro
  • Mathematics
    Electron. Notes Discret. Math.
  • 2013
Threshold functions and Poisson convergence for systems of equations in random sets
We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic progressions, sum-free sets,
k-Fold Sidon Sets
It is proved that for any integer $k \geq 1$, a -fold Sidon set A \subset [N] has at most $(N/k)^{1/2} + O((c_k^2N/ k)^{ 1/4})$ elements.


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 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
B2[g] Sets and a Conjecture of Schinzel and Schmidt
A new lower bound is obtained for F(g, n), the largest cardinality of a B2[g] set, which is proved that infn→∞, where ϵg → 0 when g → ∞.
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
Lower Bounds for Least Quadratic Non-Residues
Let p be a prime, and let n p denote the least positive integer n such that n is a quadratic non-residue mod p. In 1949, Fridlender [F] and Salie [Sa] independently showed that \( {n_p} = \Omega
Additive combinatorics
  • T. Tao, V. Vu
  • Mathematics
    Cambridge studies in advanced mathematics
  • 2007
The circle method is introduced, which is a nice application of Fourier analytic techniques to additive problems and its other applications: Vinogradov without GRH, partitions, Waring’s problem.
An introduction to additive combinatorics
This is a slightly expanded write-up of my three lectures at the Additive Combinatorics school. In the first lecture we introduce some of the basic material in Additive Combinatorics, and in the next
An Infinite Sidon Sequence
Abstract We show the existence of an infinite Sidon sequence such that the number of elements in [1,  N ] is N 2 −1+ o (1) for all large N .
A theorem in finite projective geometry and some applications to number theory
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is