No abstract

In this paper, we investigate the limiting behavior of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2, ℝ).

In this paper we explore a random process generated by the incomplete Gauss sums and establish an analogue of weak invariance principle for these sums. We focus our attention exclusively on a generalization of the limit distribution of the long incomplete Gauss sums given by the family of periodic functions analyzed by the author and Marklof.

Difference sets have wide applications in constructing sequences and codes in engineering and cryptography, and in imaging with coded masks in astronomical events and in medical events. In this paper, the relation between difference sets and binary sequences is presented and a coded mask as an application of difference sets is given. First of all, an algorithm for an appropriate difference set is written in C++ by using twin primes and then a coded mask for imaging astronomical events is designed with the aid of this algorithm.

In this paper, we investigate the distribution theorem of rational points on the closed horocycles. Sarnak proved that the long closed horocycles are equidistributed on the modular surface. We embed rational points on such horocyles on the modular surface and its tangent bundle, and study their distribution problem.