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arXiv · 1602.02360

Difference sets are not multiplicatively closed

Ilya D. Shkredov
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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.

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Ilya D. Shkredov. 2016-02-07. Difference sets are not multiplicatively closed. https://arxiv.org/abs/1602.02360

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