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1. CJM 2009 (vol 61 pp. 141)
On the Littlewood Problem Modulo a Prime 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
\[ | |A \cap (A + x)| - p/4 | = o(p).\]
Categories:42A99, 11B99 |