Minimum overlap problem

Formal statement of the problem

Let A = {a_i} and B = {b_j} be two complementary subsets, a splitting of the set of natural numbers {1, 2, …, 2n}, such that both have the same cardinality, namely n. Denote by M_k the number of solutions of the equation a_i − b_j = k, where k is an integer varying between −2n and 2n. M (n) is defined as:

The problem is to estimate M (n) when n is sufficiently large.

History

This problem can be found amongst the problems proposed by Paul Erdős in combinatorial number theory, known by English speakers as the Minimum overlap problem. It was first formulated in 1955 in the article Some remark on number theory of Riveon Lematematica, and has become one of the classical problems described by Richard K. Guy in his book Unsolved problems in number theory.

Partial results

Since it was first formulated, there has been continuous progress made in the calculation of lower bounds and upper bounds of M (n), with the following results:

Upper

J. K. Haugland showed that the limit of M (n) / n exists and that it is less than 0.385694. For his research, he was awarded a prize in a young scientists competition in 1993. In 1996, he improved the upper bound to 0.38201 using a result of Peter Swinnerton-Dyer. This has now been further improved to 0.38093.

The first known values of M(n)

The values of M (n) for the first 15 positive integers are the following:

It's just the Law of Small Numbers that it is ⌊ 5 ( n + 3 ) / 13 ⌋