Resonance Between The Representation Function And Exponential Functions Over Arithemetic Progression

Let r(n) denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x(1)(2) + x(2)(2), where x(1) and x(2) are integers. We study the behavior of the exponential sum twisted by rn over the arithmetic progressions Sigma(n equivalent to lmodq)n similar to Xr(n)e(alpha n(beta)), where 0 not equal alpha is an element of R, 0 < beta < 1, e(x) = e(2 pi ix), and n similar to X means X < n <= 2X. Here, X > 1 is a large parameter, 1 <= l <= q are integers, and (l, q) = 1. We obtain the upper bounds in different situations.