On ill-distributed $B_3$ and $B_4$ sets

André Hisatsuga, Rafael Miyazaki, Cosmin Pohoata

We show that, for $k\in \{3,4\}$, a $B_k$-set $A\subseteq [N]$ has size $o(N^{1/k})$ if it omits a fixed positive proportion of the residue classes modulo every prime in a fixed fami…