The distribution of spacings between quadratic residues

The Distribution of Spacings between Quadratic Residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution. Date: Dec 14, 1998. Sup...

The Distribution of Spacings between Quadratic Residues, Ii

We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

The Distribution of Spacings between Quadratic

The Distribution of Quadratic Residues and Non-residues

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality ...

The Distribution of Spacings between Fractional Parts of Lacunary Sequences

A primary example is to take an integer g ≥ 2 and set a(x) = g. As is true for any increasing sequence of integers, for almost every α the fractional parts αa(x) are uniformly distributed modulo 1. Moreover, for lacunary sequences, it has long been known that the fractional parts of αa(x) have strong randomness properties. For instance, the exponential sums 1 √ N ∑ x≤N cos(2παa(x)) have a Gauss...