arXiv subjects
Publications and source records attributed to E. Malavika.
In this article we prove that for a strictly increasing sequence $(a_n)$ of natural numbers, if the additive energy of $\{a_n:n\leq N\}$ is less than $N^3/(\log N)^C$ for some $C\geq14.71,$ then $(\{a_nα\})$ has Poissonian pair correlation for almost all $α\in\mathbb{R}.$ This provides a lower bound for the exponent $C$ in the additive energy bound established by Bloom and Walker [4].
Let $\mathfrak{B}$ denote the collection of odd primitive Gaussian integers and $n\mapsto b(n)$ denote the characteristic function of elements of $\mathfrak{B}$. We prove that the exponential sum $ S(α; N)=\sum_{n\le N}b(n)e(n^2α)$ satisfies \begin{equation*} \frac{S(α;N)}{N/\sqrt{\log N}} \ll N^ε(q^{-1/4}+N^{-1/2}q^{1/4}+N^{-1/8}), \end{equation*} where, $(a,q)=1$ and $|α- a/q | < 1/q^2$. Though we specialized on sums of two squares, these results extend to more general sequences.