The correlation measures of finite sequences: limiting distributions and minimum values

Abstract

Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and S\'ark\"ozy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order r. Our main result is that the correlation measure of order r for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and R\"odl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch, concerning the maximum nontrivial scalar products over a set of vectors.