### Preprints submitted for publication

#### A tight structure theorem for sumsets (with Aled Walker)

Let $$A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$$ be a finite set of non-negative integers. We prove that the sumset $$NA$$ has a certain easily-described structure, provided that $$N \geqslant b-\ell$$, as recently conjectured by Granville and Shakan. We also classify those sets $$A$$ for which this bound cannot be improved.

Article

#### Primes in short intervals: Heuristics and calculations (with Allysa Lumley)

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $$y$$ around $$x$$, where $$y\ll (\log x)^2$$. In particular we conjecture that the maximum grows surprisingly slowly as $$y$$ ranges from $$\log x$$ to $$(\log x)^2$$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification.

Article