On Diophantine problems involving fractional powers of integers

Abstract

In this thesis we are concerned with Diophantine problems of fractional degree. First we consider diagonal Diophantine inequalities of the shape

\left| \lambda_1 x_1^\theta + \cdots + \lambda_s x_s^\theta \right| < \tau,

where $\theta > 2$ is real and non-integral, $\lambda_i$ are non-zero real numbers not all of the same sign and $ \tau $ is a positive real number. For such inequalities we obtain an asymptotic formula for the number of positive integer solutions $\tuplex = (x_1, \ldots,x_s)$ inside a box of side length $P.$ Moreover, we consider the problem of representing a large positive real number by a positive definite generalised polynomial of degree $ \theta.$ Our approach follows the Davenport--Heilbronn--Freeman method. A key element in our proof is an essentially optimal mean value estimate for an exponential sum involving fractional powers of integers.

We then turn our attention to systems of simultaneous equations and inequalities. Let $\lambda_i, \mu_j$ be non-zero real numbers not all of the same sign and let $a_i, b_k$ be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape

\left| \lambda_1 x_1^\theta+ \cdots + \lambda_\ell x_\ell^\theta + \mu_1 y_1^\theta + \cdots + \mu_m y_m^\theta \right| < \tau 

a_1 x_1^d + \cdots a_\ell x_\ell^d + b_1 z_1^d + \cdots + b_n z_n^d =0,

where $ d\geq 2 $ is an integer, $ \theta > d+1$ is real and non-integral and $ \tau $ is a positive real number. For such systems we obtain an asymptotic formula for the number of positive integer solutions $(\tuplex, \tupley, \tuplez) = (x_1, \ldots, z_n)$ inside a bounded box. Our approach makes use of a two-dimensional version of the classical Hardy--Littlewood circle method and the Davenport--Heilbronn--Freeman method. The proof involves a combination of essentially optimal mean value estimates for the auxiliary exponential sums, together with estimates stemming from the classical Weyl and Weyl--van der Corput inequalities.

Date of Award	23 Mar 2021
Original language	English
Awarding Institution	University of Bristol
Supervisor	Francesco Mezzadri (Supervisor), Trevor Wooley (Supervisor) & Kevin Hughes (Supervisor)