Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Averages in vector spaces over finite fields

ANTHONY CARBERYa1, BRENDAN STONESa2 and JAMES WRIGHTa3§

a1 School of Mathematics and Maxwell Institute of Mathematical Sciences, University of Edinburgh, JCMB, King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland. e-mail: A.Carbery@ed.ac.uk

a2 The John Henry Newman School, Hitchin Road, Stevenage, Hertfordshire, G1 4AE. e-mail: brendan_maths@postmaster.co.uk

a3 School of Mathematics and Maxwell Institute of Mathematical Sciences, University of Edinburgh, JCMB, King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland. e-mail: J.R.Wright@ed.ac.uk

Abstract

We study the analogues of the problems of averages and maximal averages over a surface in $\mathbb{R}^{n}$ when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimal results in a number of model cases.

(Received August 01 2006)

(Revised April 16 2007)

Footnotes

† All authors were supported by the EC project “HARP”.

‡ Partly supported by a Leverhulme Study Abroad Fellowship and EC project “Pythagoras”.

§ Partly supported by an EPSRC grant.