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THE DENSITY OF NUMBERS REPRESENTED BY DIAGONAL FORMS OF LARGE DEGREE

Part of: Additive number theory; partitions Diophantine equations

Published online by Cambridge University Press:  23 April 2018

Brandon Hanson  and
Asif Zaman
Brandon Hanson
Affiliation:
Pennsylvania State University, University Park, PA, U.S.A. email bwh5339@psu.edu
Asif Zaman
Affiliation:
Stanford University, Stanford, CA, U.S.A. email aazaman@stanford.edu
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Abstract

Let $s\geqslant 3$ be a fixed positive integer and let $a_{1},\ldots ,a_{s}\in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form

$$\begin{eqnarray}a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}\end{eqnarray}$$
decays rapidly with respect to $k$.

MSC classification

Primary: 11D45: Counting solutions of Diophantine equations 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 542 - 550
Copyright
Copyright © University College London 2018 

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