On problems of Erdös and Rudin

A well-known conjecture of W. Rudin is that the set of squares is a ∧p-set for all p>4. In particular, this implies that for all ε>0, there exists a constant cε such that∫Π∑j=1k einj2λ4 dx14⩽cεk12+εfor any k distinct integers n1…nk. In this article we give a combinatorial interpretation of the inequality above in the spirit of ⧹|q⧹|q sum and product sets along graphs as considered by P. Erdös and E. Szemeredi (Studies in Pure Mathematics, pp. 213–218). We also show that the left-hand side of the inequality is bounded by Cεk34(log k)148−ε.