Minimum overlap problem
In number theory and set 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:
- [math]\displaystyle{ M(n) := \min_{A,B} \max_k M_k. \,\! }[/math]
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 the 1955 article Some remarks on number theory[3] (in Hebrew) in Riveon Lematematica, and has become one of the classical problems described by Richard K. 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 |
|---|---|---|
| [math]\displaystyle{ M(n) \gt n/4 }[/math] | P. Erdős | 1955 |
| [math]\displaystyle{ M(n) \gt (1 - 2^{-1/2} )\, n }[/math] | P. Erdős, Scherk | 1955 |
| [math]\displaystyle{ M(n) \gt (4 - 6^{1/2})\, n/5 }[/math] | S. Swierczkowski | 1958 |
| [math]\displaystyle{ M(n) \gt (4 - 15^{1/2})^{1/2}\, (n - 1) }[/math] | L. Moser | 1966 |
| [math]\displaystyle{ M(n) \gt (4 - 15^{1/2})^{1/2}\, n }[/math] | J. K. Haugland | 1996 |
| [math]\displaystyle{ M(n) \gt 0.379005 { \, } n }[/math] | E. P. White | 2022 |
Upper
| Limit superior | Author(s) | Year |
|---|---|---|
| [math]\displaystyle{ M(n) \lt (1+o(1)) n/2 }[/math] | P. Erdős | 1955 |
| [math]\displaystyle{ M(n) \lt (1+o(1)) 2n/5 }[/math] | T. S. Motzkin, K. E. Ralston and J. L. Selfridge, | 1956 |
| [math]\displaystyle{ M(n) \lt (1+o(1)) 0.382002... n }[/math] | J. K. Haugland | 1996 |
| [math]\displaystyle{ M(n) \lt (1+o(1)) 0.380926... n }[/math] | J. K. Haugland | 2016 |
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.[4] In 1996, he improved the upper bound to 0.38201 using a result of Peter Swinnerton-Dyer.[5][2] This has now been further improved to 0.38093.[6] In 2022, the lower bound was shown to be at least 0.379005 by E. P. White.[7]
The first known values of M(n)
The values of M (n) for the first 15 positive integers are the following:[1]
| [math]\displaystyle{ n\,\! }[/math] | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ... |
| [math]\displaystyle{ M(n)\,\! }[/math] | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | ... |
It is just the Law of Small Numbers that it is [math]\displaystyle{ \textstyle \lfloor 5(n+3)/13 \rfloor }[/math][1]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 Guy, Richard K. (2004). "C17". in Bencsáth, Katalin A.; Halmos, Paul R.. Unsolved Problems in Number Theory. New York: Springer Science+Business Media Inc.. pp. 199–200. ISBN 0-387-20860-7.
- ↑ 2.0 2.1 2.2 2.3 Finch, Steven (2 July 2004). "Erdös' minimum overlap problem". Archived from the original on 5 April 2015. https://web.archive.org/web/20150405060955/http://www.people.fas.harvard.edu/~sfinch/csolve/ovr.pdf. Retrieved 15 December 2013.
- ↑ P. Erdős: Some remarks on number theory (in Hebrew), Riveon Lematematika 9 (1955), 45-48 MR17,460d.
- ↑ Haugland, Jan Kristian. "The minimum overlap problem". http://www.neutreeko.net/mop/index.htm. Retrieved 20 September 2016.
- ↑ Haugland, Jan Kristian (1996). "Advances in the Minimum Overlap Problem". Journal of Number Theory (Ohio (USA)) 58 (1): 71–78. doi:10.1006/jnth.1996.0064. ISSN 0022-314X.
- ↑ Haugland, Jan Kristian (2016). "The minimum overlap problem revisited". arXiv:1609.08000 [math.GM].
- ↑ White, Ethan Patrick (2022). "Erdős' minimum overlap problem". arXiv:2201.05704 [math.CO].
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