On the Littlewood Problem Modulo a Prime

http://dx.doi.org/10.4153/CJM-2009-007-4
Canad. J. Math. 61(2009), 141-164

  Published:2009-02-01
 Printed: Feb 2009

Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow \mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f} \Vert_1 \leq 1$. Then $\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$. 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 $A \subseteq \Zp$ is a set of cardinality $\lfloor p/2\rfloor$, then $\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This gives a result on a ``mod $p$'' 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 $A \subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is some $x \in \Zp$ such that \[ | |A \cap (A + x)| - p/4 | = o(p).\]