Sidon sequence

In number theory, a Sidon sequence (or Sidon set), named after the Hungarian mathematician Simon Sidon, is a sequence A = {a0, a1, a2, ...} of natural numbers in which all pairwise sums ai + aj (i ≤ j) are different. Sidon introduced the concept in his investigations of Fourier series. In number theory, a Sidon sequence (or Sidon set), named after the Hungarian mathematician Simon Sidon, is a sequence A = {a0, a1, a2, ...} 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, 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, the question remained unsolved for almost 80 years. In 2010, it was finally settled by J. Cilleruelo, I. Ruzsa and C. Vinuesa. 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 x + O ( x 4 ) {displaystyle {sqrt {x}}+O({sqrt{x}})} . Using a construction of J. Singer, they showed that there exist Sidon sequences that contain x ( 1 − o ( 1 ) ) {displaystyle {sqrt {x}}(1-o(1))} terms less than x. 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