arXiv subjects
Publications and source records attributed to Kirsti Biggs.
A result of Wright from 1937 shows that there are arbitrarily large natural numbers which cannot be represented as sums of $s$ $k$th powers of natural numbers which are constrained to lie within a narrow region. We show that the analogue of this result holds in the shifted version of Waring's problem.
The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number $α$ when the base field $\mathbb{K}$ is a number field and $\mathbb{K}(α)/\mathbb{K}$ is Galois. Our second result establishes an explicit height bound for any non-zero element $α$ which is not a root of unity in a Galois extension $\mathbb{F}/\mathbb{K}$, depending on the degree of $\mathbb{K}/\mathbb{Q}$ and the number of conjugates of $α$ which are multiplicatively independent over $\mathbb{K}$. As a consequence, we obtain a height bound for such $α$ that is independent of the multiplicative independence condition.
We show that for integers $k\geq 4$ and $s\geq k^2+(3k-1)/4$, we have an asymptotic formula for the number of solutions, in positive integers $x_i$, to the inequality $\left|(x_1-θ_1)^k+\dotsc+(x_s-θ_s)^k-τ\right|<η$, where $θ_i\in(0,1)$ with $θ_1$ irrational, $η\in(0,1]$, and $τ>0$ is sufficiently large. We use Freeman's variant of the Davenport--Heilbronn method, along with a new estimate on the Hardy--Littlewood minor arcs, to obtain this improvement on the original result of Chow.