Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Congruences involving product of intervals and sets with small
  multiplicative doubling modulo a prime and applications   In the present paper we obtain new upper bound estimates for the number of
solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad
x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$
is a subset of the field of residue classes modulo $p$ having small
multiplicative doubling. We then use this estimate to show that the number of
solutions of the congruence $$ x^n\equiv \lambda\pmod p; \quad x\in \N, \quad
L<x<L+p/n, $$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers
$n, \lambda$ and $L$, for some absolute constant $c>0$. This implies, in
particular, that if $f(x)\in \Z[x]$ is a fixed polynomial without multiple
roots in $\C$, then the congruence $ x^{f(x)}\equiv 1\pmod p, \,x\in
\mathbb{N}, \,x\le p,$ has at most $p^{\frac{1}{3}-c}$ solutions as
$p\to\infty$, improving some recent results of Kurlberg, Luca and Shparlinski
and of Balog, Broughan and Shparlinski. We use our results to show that almost
all the residue classes modulo $p$ can be represented in the form $xg^y \pmod
p$ with positive integers $x<p^{5/8+\varepsilon}$ and $y<p^{3/8}$. Here $g$
denotes a primitive root modulo $p$. We also prove that almost all the residue
classes modulo $p$ can be represented in the form $xyzg^t \pmod p$ with
positive integers $x,y,z,t<p^{1/4+\varepsilon}$.

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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg