Author:
Hegyvári Norbert,Hennecart François
Abstract
Abstract
In the last decade the growth in groups focused the interest of many researchers in the domain of additive combinatorics.
In this note, we investigate the special case of the Heisenberg group
{\mathcal{H}}
of order 3 on different fields R, namely,
{R=\mathbf{R}}
and
{R=\mathbf{F}_{p}}
.
Denoting by
{[x,y,z]}
the elements of
{\mathcal{H}}
, a cube is a subset of the type
{[A,B,C]}
, with
{A,B,C\subset R}
.
This study relies to the well-known sum-product problem and its extensions, from which we obtain strong lower bounds for the size of the square
{[A,B,C]^{2}}
of cubes in
{\mathcal{H}}
.
In the final section, we also consider the problem of counting the subsets of
{\mathcal{H}}
of the type
{[A,B,C]^{2}}
.
Subject
Applied Mathematics,General Mathematics
Reference34 articles.
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3. On the number of incidences between planes and points in three dimensions;Preprint,2014
4. Variations on the sum-product problem;SIAM J. Discrete Math.,2015
5. New sum-product type estimates over finite fields;Adv. Math.,2016
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