Sums of two squares in short intervals

Let S denote the set of integers representable as a sum of two squares. Since S can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that S has many properties in common with the set of prime numbers. In this paper we exhibit "unexpected irregularities" in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of S than expected, and infinitely many intervals containing considerably fewer than expected.