A note on the set $\boldsymbol{A(A+A)}$
Abstract
Let $p$ a large enough prime number. When $A$ is a subset of $\mathbb{F}_p\smallsetminus\{0\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbb{F}_p\smallsetminus\{0\}\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\ge 0.3051 p$ implies $A(A+A)\supseteq\mathbb{F}_p\smallsetminus\{0\}$. In the opposite direction we show that there exists a set $A$ such that $|A| > (1/8+o(1))p$ and $\mathbb{F}_p\smallsetminus\{0\}\not\subseteq A(A+A)$.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.08869
- arXiv:
- arXiv:1811.08869
- Bibcode:
- 2018arXiv181108869B
- Keywords:
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- Mathematics - Number Theory;
- 11B75
- E-Print:
- 10 pages