On the distribution of runners on a circle

Consider $n$ runners running on a circular track of unit length with constant speeds such that $k$ of the speeds are distinct. We show that, at some time, there will exist a sector $S$ which contains at least $|S|n+ Ω(\sqrt{k})$ runners. The result can be generalized as follows. Let $f(x,y)$ be a complex bivariate polynomial whose Newton polytope has $k$ vertices. Then there exists $a\in {\mathbb C}\setminus\{0\}$ and a complex sector $S=\{re^{\imath θ}: r>0, α\leq θ\leq β\}$ such that the univariate polynomial $f(x,a)$ contains at least $\frac{β-α}{2π}n+Ω(\sqrt{k})$ non-zero roots in $S$ (where $n$ is the total number of such roots and $0\leq (β-α)\leq 2π$). This shows that the Real $τ$-Conjecture of Koiran implies the conjecture on Newton polytopes of Koiran et al.

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