3 problems

Let $k$ and $\ell$ be multiplicatively independent positive integers. A subset of $\mathbb{N}$ is sparse if its counting function is bounded by $O((\log x)^d)$ for some $d$. Let…

For real $x$ and $t>1$, let $S_{\left\lfloor x/n^t\right\rfloor,x}$ denote the set considered in the paper, with the function defining $S_{f,x}$ taken to be non-piecewise. Sparsity…

Let $A\subset[n]$ be a set whose density is $|A|/N$, where $N=n$. Fix an integer $k\geq 2$, and consider the configuration of two points $x$ and $x+d^k$ with $d\neq 0$ an integer.…