On sum-product representations in ${\mathbb{Z}}_q$

Chang, Mei-Chu. "On sum-product representations in $\mathbb {Z}_q$." Journal of the European Mathematical Society 008.3 (2006): 435-463. <http://eudml.org/doc/277412>.

@article{Chang2006,
abstract = {The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of $\mathbb \{Z\}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $|A|\sim \log q$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll \log q$ and $\log k\ll \log q$, provided the residue sets $\pi _\{q^\{\prime \}\}(A)$ are large for all large divisors $q^\{\prime \}$ of $q$. Even in the special case of prime modulus $q$, some results are new, when considering large but bounded sets $A$. It follows for instance from our estimates that one can obtain $r$ as small as $r\sim \log q/\log |A|$ with similar restriction on $k$, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman’s structural theorem on sets with small doubling constant. Taking for $A=H$ a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on $\min _\{a\in \mathbb \{Z\}^*_q\}\max _\{x\in H\}\Vert ax/q\Vert $ are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of $x^m=a~\@mod \;q$ to composite modulus $q$.},
author = {Chang, Mei-Chu},
journal = {Journal of the European Mathematical Society},
language = {eng},
number = {3},
pages = {435-463},
publisher = {European Mathematical Society Publishing House},
title = {On sum-product representations in $\mathbb \{Z\}_q$},
url = {http://eudml.org/doc/277412},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Chang, Mei-Chu
TI - On sum-product representations in $\mathbb {Z}_q$
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 3
SP - 435
EP - 463
AB - The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of $\mathbb {Z}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $|A|\sim \log q$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll \log q$ and $\log k\ll \log q$, provided the residue sets $\pi _{q^{\prime }}(A)$ are large for all large divisors $q^{\prime }$ of $q$. Even in the special case of prime modulus $q$, some results are new, when considering large but bounded sets $A$. It follows for instance from our estimates that one can obtain $r$ as small as $r\sim \log q/\log |A|$ with similar restriction on $k$, something not covered by earlier work of Konyagin and Shparlinski. On the technical side, essential use is made of Freiman’s structural theorem on sets with small doubling constant. Taking for $A=H$ a possibly very small multiplicative subgroup, bounds on exponential sums and lower bounds on $\min _{a\in \mathbb {Z}^*_q}\max _{x\in H}\Vert ax/q\Vert $ are obtained. This is an extension to the results obtained by Konyagin, Shparlinski and Robinson on the distribution of solutions of $x^m=a~\@mod \;q$ to composite modulus $q$.
LA - eng
UR - http://eudml.org/doc/277412
ER -