Finite-set approximation of fractional-dilate sum-dilate data

Let $\alpha$ be a fractional dilate, let $N$ be a positive integer, and let $\epsilon>0$.

Finite-set approximation conjecture. There exists a finite subset $S$ of the integers such that, for every integer $k$ with $|k|\leq N$,

$$\left|\frac{\log\norm{\alpha+k\cdot\alpha}}{\log\norm\alpha}-\frac{\log|S+k\cdot S|}{\log|S|}\right|<\epsilon.$$

This gives a finite-set approximation to the normalized sum-dilate quantities of a fractional dilate, uniformly for the bounded range $|k|\leq N$. The statement is presented among the paper's open questions and is therefore open.