Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

Let f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let {lambda(f) (n)} (n) be its sequence of normalised Fourier coefficients. We show that if K/Q is any number field, and N K denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers n. N (K) yield a sign change for the sequence {lambda(f) (n)} (n epsilon NK.) More precisely, for a positive proportion of n epsilon N (K) n [1, X] we have lambda(f) (n)lambda(f) (n') < 0, where n' is the first element of N (K) greater than n for which lambda(f) (n') not equal 0. For example, for K = Q(i) and N (K) = {m(2) + n(2) : m, n epsilon Z} the set of sums of two squares, we obtain >> (f) X/root log X such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomaki and Radziwill on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed a not equal 0 there are >> f, epsilon X1/2- epsilon sign changes for lambda(f) along the sequence of integers of the form a+ m(2) + n(2) <= X. 2020 Mathematics Subject Classification: 11N37, 11N64, 11F30 (Primary); 11F11, 11N36, 11R47 (Secondary)