Erdős-Heilbronn conjecture
Let $A\subset {\mathbb{Z}}_{p}$ be a set of residues modulo $p$, and let $h$ be a positive integer, then
$${h}^{\wedge}A=\{{a}_{1}+{a}_{2}+\mathrm{\cdots}+{a}_{h}\mid {a}_{1},{a}_{2},\mathrm{\dots},{a}_{h}\text{are distinct elements of}A\}$$ |
has cardinality at least $\mathrm{min}(p,hk-{h}^{2}+1)$. This was conjectured by Erdős and Heilbronn in 1964[1]. The first proof was given by Dias da Silva and Hamidoune in 1994.
References
- 1 Paul Erdős and Hans Heilbronn. On the addition of residue classes^{} $modp$. Acta Arith., 9:149–159, 1964. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0156.04801Zbl 0156.04801.
- 2 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.
Title | Erdős-Heilbronn conjecture |
---|---|
Canonical name | ErdHosHeilbronnConjecture |
Date of creation | 2013-03-22 13:38:15 |
Last modified on | 2013-03-22 13:38:15 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 10 |
Author | bbukh (348) |
Entry type | Theorem |
Classification | msc 11B75 |
Related topic | CauchyDavenportTheorem |