From small eigenvalues to large cuts, and Chowla's cosine problem

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|\lambda_n|\leq d^{\gamma}$ contains a clique of size $d^{1-O(\gamma)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine problem (1965): for every finite set $A\subseteq \mathbb{Z}_{>0}$, the minimum of the cosine polynomial satisfies $$\min_{x\in [0, 2\pi]}\sum_{a\in A}\cos(ax)\leq -|A|^{1/10-o(1)}.$$ Another application makes significant progress on the problem of MaxCut in $H$-free graphs initiated by Erd\H{o}s and Lov\'asz in the 1970's. We show that every $m$-edge graph with no clique of size $m^{1/2-\delta}$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(\delta)>0$.