Sonar signal processing
In number theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955.[1][2]
Formal statement of the problem
Let A = {ai} and B = {bj} 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 Mk the number of solutions of the equation ai − bj = 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.[2]
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 Guy in his book Unsolved problems in number theory.[1]
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:[1][2]
Lower
| Limit inferior | Author(s) | Year |
|---|---|---|
| P. Erdős | 1955 | |
| P. Erdős, Scherk | 1955 | |
| S. Swierczkowski | 1958 | |
| L. Moser | 1966 | |
| J. K. Haugland | 1996 |
Upper
| Limit superior | Author(s) | Year |
|---|---|---|
| P. Erdős | 1955 | |
| T. S. Motzkin, K. E. Ralston and J. L. Selfridge, | 1956 | |
| J. K. Haugland | 1996 |
J. K. Haugland showed that the limit of M(n)/n < 0.38569401 unconditionally. For his research, he was awarded a prize in the competition for young scientists in 1993.[3] In 1996 he showed that the superior and inferior limits of M(n)/n are same, thus proving that the limit of M(n)/n exists.[2][4]
The first known values of M(n)
The values of M(n) for the first 15 positive integers are the following:[1]
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ... |
| M(n) | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | ... |
References
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