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 nonintegral, $\lambda_i$ are nonzero 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 DavenportHeilbronnFreeman 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 nonzero real numbers not all of the same sign and let $a_i, b_k$ be nonzero 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 nonintegral 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 twodimensional version of the classical HardyLittlewood circle method and the DavenportHeilbronnFreeman 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 Weylvan der Corput inequalities.
Date of Award  23 Mar 2021 

Original language  English 
Awarding Institution 

Supervisor  Francesco Mezzadri (Supervisor), Trevor Wooley (Supervisor) & Kevin Hughes (Supervisor) 