Title:

On additive problems involving shifted integers and ellipsephic sets

In this thesis, we present a series of results concerning the number of solutions to equations and inequalities involving sums of perfect powers. In order to count such solutions, we use variants of the HardyLittlewood circle method, a versatile technique whose history is outlined in Chapter 1. Firstly, to handle inequalities we use the DavenportHeilbronn version of the circle method, in the form developed by Freeman, which we provide an introduction to in Chapter 2. In Chapter 3, we apply this method to the problem of counting solutions to an inequality in which our variables have been shifted by small real numbers. Speciﬁcally, for natural numbers k and s and real numbers θ1,...,θs ∈ (0,1) with θ1 irrational, let N(τ) be the number of solutions in positive integers xi to the inequality I (x1 −θ1)k + ... + (xs −θs)k −τ I < 1. We show that an asymptotic formula for N(τ) holds whenever k ≥ 4 ands ≥ k2 + (3k −1)/4, an improvement on a result of Chow. We also prove a shifted analogue of a result of Wright showing that such an inequality does not always have solutions in which the variables are forced to lie in a short interval. We then turn to problems involving integers whose digits in a given base are restricted to certain values; we refer to such integers as ellipsephic. We use Wooley's eﬃcient congruencing method to bound the number of ellipsephic solutions to the Vinogradov system xj 1 + ... + xj s = yj 1 + ... + yj s, (1 ≤ j ≤ k), handling the case k = 2 in Chapter 4, and the general case in Chapter 5. The additive structure of our ellipsephic sets enables us to achieve better bounds than those previously available, since any application of earlier results would use only the density of the set of variables within the natural numbers. For example, in the case where our variables have square digits, we obtain diagonal behaviour with twice as many variables as in the classical case.
