Sidon sequence

In number theory, a Sidon sequence (or Sidon set), named after the Hungarian mathematician Simon Sidon, is a sequence A = {a0a1a2, ...} of natural numbers in which all pairwise sums ai + aj (i ≤ j) are different. Sidon introduced the concept in his investigations of Fourier series.

The main problem in the study of Sidon sequences, posed by Sidon,[1] is to find the largest number of elements a Sidon sequence A can have smaller than some given number x. Despite a large body of research,[2] the question remained unsolved for almost 80 years. In 2010, it was finally settled[3] by J. Cilleruelo, I. Ruzsa and C. Vinuesa.

Early results

Paul Erdős and Pál Turán proved that, for every x > 0, the number of elements smaller than x in a Sidon sequence is at most $\sqrt{x}+O(\sqrt[4]{x})$. Using a construction of J. Singer, they showed that there exist Sidon sequences that contain $\sqrt{x}(1-o(1))$ terms less than x.

Infinite Sidon sequences

Erdős also showed that if we consider any particular infinite Sidon sequence A and let A(x) denote the number of its elements up to x, then

$\liminf_{x \to \infty} \frac{A(x)\sqrt{\log x}}{\sqrt{x}}\leq 1$.

That is, infinite Sidon sequences are thinner than the densest finite Sidon sequences.

For the other direction, Chowla and Mian observed that the greedy algorithm gives an infinite Sidon sequence with $A(x)>c\sqrt[3]{x}$ for every x.[4] Ajtai, Komlós, and Szemerédi improved this with a construction[5] of a Sidon sequence with

$A(x)>\sqrt[3]{x\log x}.$

The best lower bound to date was given by Imre Z. Ruzsa, who proved[6] that a Sidon sequence with

$A(x)>x^{\sqrt{2}-1-o(1)}$

exists. Erdős conjectured that an infinite Sidon set A exists for which $A(x)>x^{1/2-o(1)}$ holds. He and Rényi showed[7] the existence of a sequence {a0,a1,...} with the conjectural density but satisfying only the weaker property that there is a constant k such that for every natural number n there are at most k solutions of the equation ai + aj = n. (To be a Sidon sequence would require that k = 1.)

Erdős further conjectured that there exists a nonconstant integer-coefficient polynomial whose values at the natural numbers form a Sidon sequence. Specifically, he asked if the set of fifth powers is a Sidon set. Ruzsa came close to this by showing that there is a real number c with 0 < c < 1 such that the range of the function f(x) = x5 + [cx4] is a Sidon sequence, where [.] denotes integer part. As c is irrational, this function f(x) is not a polynomial. The statement that the set of fifth powers is a Sidon set is a special case of the later conjecture of Lander, Parkin and Selfridge.

Relationship to Golomb rulers

All finite Sidon sets are Golomb rulers, and vice-versa.

To see this, suppose for a contradiction that S is a Sidon set and not a Golomb ruler. Since it is not a Golomb ruler, there must be four members such that $a_i-a_j=a_k-a_l$. It follows that $a_i+a_l=a_k+a_j$, which contradicts the proposition that S is a Sidon set. Therefore all Sidon sets must be Golomb rulers. By a similar argument, all Golomb rulers must be Sidon sets.