Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most exponentially in $N$ (for fixed $k$). Moreover it is known that the decay is at worst polynomial if $k < 5$. But no general sub-exponential bound is known if $k \geq 5$.
In this talk I will present evidence that the modulus decreases at most polynomially for prime values of $N$ by showing that counterexamples must be very sparse. We do this by counting rational points that approximate a set that is definable in an o-minimal structure. This is motivated by the counting results of Bombieri-Pila and Pila-Wilkie.
I will also discuss progress on Myerson's related conjecture on Gaussian periods, as well as strong equidistribution properties of tuples of roots of unity, and connections to an ergodic result of Lind-Schmidt-Verbitskiy.