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On the Littlewood Problem Modulo a Prime

Part of: Sequences and sets Harmonic analysis in one variable

Published online by Cambridge University Press:  20 November 2018

Ben Green  and
Sergei Konyagin
Ben Green
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom, b.j.green@dpmms.cam.ac.uk
Sergei Konyagin
Affiliation:
Department of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia, konyagin@ok.ru
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Abstract

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Let $p$ be a prime, and let $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{R}$ be a function with $\mathbb{E}f=0$ and $||\hat{f}|{{|}_{1}}\le 1$. Then ${{\min }_{x\in \mathbb{Z}/p\mathbb{Z}}}|f\left( x \right)|=O{{\left( \log p \right)}^{-1/3+\in }}$. One should think of $f$ as being “approximately continuous”; our result is then an “approximate intermediate value theorem”.

As an immediate consequence we show that if $A\subseteq \mathbb{Z}/p\mathbb{Z}$ is a set of cardinality $\left\lfloor {p}/{2}\; \right\rfloor $, then ${{\sum }_{r}}\widehat{|\,{{1}_{A}}}\left( r \right)|\gg {{\left( \log p \right)}^{1/3-\in }}$. This gives a result on a “$\,\bmod \,p$” analogue of Littlewood's well-known problem concerning the smallest possible ${{L}^{1}}$-norm of the Fourier transform of a set of $n$ integers.

Another application is to answer a question of Gowers. If $A\,\subseteq \,{\mathbb{Z}}/{p\mathbb{Z}}\;$ is a set of size $\left\lfloor {p}/{2}\; \right\rfloor $, then there is some $x\,\in \,\mathbb{Z}/p\mathbb{Z}$ such that

$$||A\cap \left( A+x \right)\,-\,p/4|\,=o\left( p \right).$$

MSC classification

Primary: 42A99: None of the above, but in this section
Secondary: 11B99: None of the above, but in this section
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 1 , 01 February 2009 , pp. 141 - 164
Copyright
Copyright © Canadian Mathematical Society 2009

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