Mathematics Colloquia and Seminars

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Description

Waring's problem (in its modern form) is concerned with determining, for each $2\leq k\in \mathbb{N}$, the smallest integer $G(k)$ such that for any fixed $s\geq G(k)$ there always exist non-negative integer solutions $(x_1,\ldots,x_s)$ to the equation 

$$x_1^k+\cdots+x_s^k=N$$

for every sufficiently large $N\in \mathbb{N}$. We explore variants of this problem when the integer $k$ is replaced with a non-integer $\theta$, and compare the main differences that arise when working in this setting. For example, we examine non-negative integer solutions $(x_1,\ldots,x_s)$ to the equation

$\lfloor x_1^{\theta}\rfloor+\cdots+\lfloor x_s^{\theta}\rfloor=N$

for large $N\in \mathbb{N}$ and reduce the number of variables required to establish an asymptotic formula for the number of solutions.