arXiv subjects
Publications and source records attributed to Constantinos Poulias.
Let $θ>2$ be real and non-integral with integer part $n = \lfloor θ\rfloor$ and let $ ϕ(x)$ be a generalised polynomial with leading term $x^θ.$ We establish a mean value estimate for the exponential sum \begin{equation*} \sum_{1 \leq x \leq P} e \left(α_1 x + \cdots + α_n x^n + α_ϕϕ(x) \right). \end{equation*}
This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ θ,$ where $ θ>2$ is real and non-integral. For fixed non-zero real numbers $ λ_i $ not all of the same sign we write \begin{equation*} \mathcal F (\textbf{x}) = λ_1 x_1^θ+ \cdots + λ_s x_s^θ. \end{equation*} For a fixed positive real number $ τ$ we give an asymptotic formula for the number of positive integer solutions of the inequality $ | \mathcal F (\textbf{x}) | < τ$ inside a box of side length $P.$ Moreover, we investigate the problem of representing a large positive real number by a positive definite generalized polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.
Let $λ_i, μ_j$ 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 \begin{equation*} \begin{cases} \left| λ_1 x_1^θ+ \cdots + λ_\ell x_\ell^θ+ μ_1 y_1^θ+ \cdots + μ_m y_m^θ\right| < τ\\[10pt] a_1 x_1^d + \cdots a_\ell x_\ell^d + b_1 z_1^d + \cdots + b_n z_n^d =0, \end{cases} \end{equation*} where $ d\geq 2 $ is an integer, $ θ> d+1$ is real and non-integral and $ τ$ is a positive real number. For such systems we obtain an asymptotic formula for the number of positive integer solutions $(\textbf{x}, \textbf{y}, \textbf{z}) = (x_1, \ldots, z_n)$ 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.