On the Erdős–Turán conjecture

TextLet NN be the set of all nonnegative integers and k≥2k≥2 be a fixed integer. For a set A⊆NA⊆N, let rk(A,n)rk(A,n) denote the number of solutions of a1+⋯+ak=na1+⋯+ak=n with a1,…,ak∈Aa1,…,ak∈A. In this paper, we prove that for given positive integer u  , there is a set A⊆NA⊆N such that rk(A,n)≥1rk(A,n)≥1 for all n≥0n≥0 and the set of n   with rk(A,n)=k!urk(A,n)=k!u has density one. This generalizes recent results of Chen and Yang.VideoFor a video summary of this paper, please visit http://youtu.be/2fbKtDAOqQ0.