Difference sets are not multiplicatively closed
Number Theory
2016-10-04 v4 Combinatorics
Abstract
We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. A similar result takes place in the prime field F_p for sufficiently small D. It gives, in particular, that multiplicative subgroups of size less than p^{4/5-\eps} cannot be represented in the form A-A for any A from F_p.
Cite
@article{arxiv.1602.02360,
title = {Difference sets are not multiplicatively closed},
author = {Ilya D. Shkredov},
journal= {arXiv preprint arXiv:1602.02360},
year = {2016}
}
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21 pages