Montreal, December 4 - 7, 2020
Granville--Shakan recently showed that, if $N$ is large enough, the answer to this question is yes, equality does hold. However, for all $d \geqslant 2$ their results gave only an ineffective lower bound on what 'large enough' should mean. In this talk we will describe two new pieces of work on this question: a new bound in the case $d=1$, which is tight for several infinite families of sets $A$, and the first effective bounds for arbitrary $A$ when $d \geqslant 2$. These results are joint work with Granville and with Granville--Shakan respectively. If time permits, we will describe the connections between this work and Khovanskii's theorem (that the size of $NA$ is a polynomial in $N$, for large enough $N$).