Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Littlewood problem modulo a prime   Let p be a prime, and let f : Z/pZ -> R be a function with average value 0
and ||f||_A <= 1, where ||f||_A denotes the algebra norm (L^1 norm of the
Fourier transform). Then f(x) is small for some x, specifically min_x |f(x)| is
no more than O(log p)^{-1/3 + eps}. 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 B in Z/pZ is a set of cardinality
(p-1)/2 then the algebra norm ||1_B||_A is >> (log p)^{1/3 - eps}. 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 B in Z/pZ is a set
of size (p-1)/2 then there is some x in Z/pZ such that the intersection of B
with B + x has cardinality within o(p) of p/4.

8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con