On the size of the set A(A+1)
| dc.creator | Garaev, M. Z. | |
| dc.creator | Shen, Chun-Yen | |
| dc.date | 2008-11-26 | |
| dc.date | 2008-12-16 | |
| dc.date.accessioned | 2026-07-07T12:12:38Z | |
| dc.date.available | 2026-07-07T12:12:38Z | |
| dc.description | Let $F_p$ be the field of a prime order $p.$ For a subset $A\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p.$ In the present paper we show that if $|A|<p^{1/2},$ then $$ |A(A+1)|\ge |A|^{106/105+o(1)}.$$ If $|A|>p^{2/3},$ then we prove that $$|A(A+1)|\gg \sqrt{p |A|}$$ and show that this is the optimal in general settings bound up to the implied constant. We also estimate the cardinality of $A(A+1)$ when $A$ is a subset of real numbers. We show that in this case one has the Elekes type bound $$ |A(A+1)|\gg |A|^{5/4}. $$ | |
| dc.description | Minor corrections | |
| dc.identifier | https://arxiv.org/abs/0811.4206 | |
| dc.identifier | http://arxiv.org/abs/0811.4206 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/210608 | |
| dc.subject | Number Theory | |
| dc.subject | 11B75 | |
| dc.title | On the size of the set A(A+1) | |
| dc.type | text |