Talk

Let k be a natural number and A be a subset of the natural numbers. Can one determine the lower exponential density of the set
$$\{ x^k + a:  a\in A, x\in\mathbb N \}$$
in terms of lower exponential density of A? This problem is trivial for k=1, and has an elementary complete solution when k=2. Eighty years ago Davenport gave lower bounds in all cases of interest. In this talk we describe joint work with Simon Myerson. Davenport's estimates are improved when A is already fairly dense. In some range, our results are best possible.

For $\ell$ an odd prime number and $d$ a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$ due to Ellenberg—Venkatesh. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$.