The asymptotic sum-dilate formula for fractional dilates

Let a fractional dilate be an object

for which $ orm{}>1$, let $k$ be a positive integer, and let $A_n$ be drawn from $^n$. **The asymptotic \sum-dilate conjecture.**

\lim_{n\to\infty}\frac{\log |A_n+k\cdot A_n|}{n}=\log\norm{+k\cdot}.

This would extend results about sizes of sums and differences of finite sets to fractional dilates; for example, Ruzsa's triangle inequality would imply $\norm{}\norm{\beta-\gamma}\leq\norm{-\beta}\norm{-\gamma}$ for fractional dilates

, $\beta$, and $\gamma$. The source presents this as an open question.