Abstract view
On the Littlewood Problem Modulo a Prime


Published:20090201
Printed: Feb 2009
Ben Green
Sergei Konyagin
Abstract
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 wellknown
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
\[  A \cap (A + x)  p/4  = o(p).\]