In number theory, a Sidon sequence is a sequence of natural numbers in which all pairwise sums (for ) are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian mathematician Simon Sidon, who 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 maximum number of elements that a Sidon sequence can contain, up to some bound . Despite a large body of research,[2] the question has remained unsolved.[3]
Paul Erdős and Pál Turán proved that, for every , the number of elements smaller than in a Sidon sequence is at most . Several years earlier, James Singer had constructed Sidon sequences with terms less than x. The bound was improved to in 1969[4] and to in 2023.[5]
In 1994 Erdős offered 500 dollars for a proof or disproof of the bound .[6]
Erdős also showed that, for any particular infinite Sidon sequence with denoting the number of its elements up to ,
For the other direction, Chowla and Mian observed that the greedy algorithm gives an infinite Sidon sequence with for every .[7] Ajtai, Komlós, and Szemerédi improved this with a construction[8] of a Sidon sequence with
The best lower bound to date was given by Imre Z. Ruzsa, who proved[9] that a Sidon sequence with
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 with such that the range of the function is a Sidon sequence, where denotes the integer part. As is irrational, this function 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.
The existence of Sidon sequences that form an asymptotic basis of order (meaning that every sufficiently large natural number can be written as the sum of numbers from the sequence) has been proved for in 2010,[11] in 2014,[12] (the sum of four terms with one smaller than , for arbitrarily small positive ) in 2015[13] and in 2023 as a preprint,[14][15] this later one was posed as a problem in a paper of Erdős, Sárközy and Sós in 1994.[16]
All finite Sidon sets are Golomb rulers, and vice versa.
To see this, suppose for a contradiction that is a Sidon set and not a Golomb ruler. Since it is not a Golomb ruler, there must be four members such that . It follows that , which contradicts the proposition that is a Sidon set. Therefore all Sidon sets must be Golomb rulers. By a similar argument, all Golomb rulers must be Sidon sets.