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SOLUTIONS TO DIAGONAL CONGRUENCES WITH VARIABLES RESTRICTED TO A BOX

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  03 April 2018

Todd Cochrane ,
Misty Ostergaard  and
Craig Spencer
Todd Cochrane
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. email cochrane@math.ksu.edu
Misty Ostergaard
Affiliation:
Department of Mathematics, University of Southern Indiana, Evansville, IN 47712, U.S.A. email m.ostergaard@usi.edu
Craig Spencer
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A. email cvs@math.ksu.edu
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Abstract

We prove that for any positive integers $k,n$ with $n>\frac{3}{2}(k^{2}+k+2)$, prime $p$, and integers $c,a_{i}$, with $p\nmid a_{i}$, $1\leqslant i\leqslant n$, there exists a solution $\text{}\underline{x}$ to the congruence

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}a_{i}x_{i}^{k}\equiv c\hspace{0.6em}({\rm mod}\hspace{0.2em}p)\end{eqnarray}$$
with $1\leqslant {x_{i}\ll }_{k}p^{1/k}$, $1\leqslant i\leqslant n$. This upper bound is best possible. Refinements are given for smaller $n$, and for variables restricted to intervals in more general position. In particular, for any $\unicode[STIX]{x1D700}>0$ we give an explicit constant $c_{\unicode[STIX]{x1D700}}$ such that if $n>c_{\unicode[STIX]{x1D700}}k$, then there is a solution with $1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$.

MSC classification

Primary: 11L07: Estimates on exponential sums 11L03: Trigonometric and exponential sums, general
Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 430 - 444
Copyright
Copyright © University College London 2018 

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