Mixing and asymptotic distribution modulo 1

Distribution of Powers modulo 1 and Relatedtopicsmiguel

This is a review of several results related to distribution of powers and combinations of powers modulo 1. We include a proof that given any sequence of real numbers n , it is possible to get an (given 6 = 0), or a (given > 1) such that n is close to n modulo 1. We also prove that in a number eld, if a combination of powers 1 n 1 + + m n m has bounded v-adic absolute value (where v is any non-A...

Asymptotic Distribution of Divergence Measure with Applications

Distribution of Residues Modulo p

The distribution of quadratic residues and non-residues modulo p has been of intrigue to the number theorists of the last several decades. Although Gauss’ celebrated Quadratic Reciprocity Law gives a beautiful criterion to decide whether a given number is a quadratic residue modulo p or not, it is still an open problem to find a small upper bound on the least quadratic non-residue mod p as a fu...

On uniform distribution modulo one

We introduce an elementary argument to the theory of distribution of sequences modulo one.

Distribution modulo One and Ratner’s Theorem

Contents 1. Introduction 1 2. Randomness of point sequences mod 1 2 2.1. Distribution of gaps 4 2.2. Independent random variables 6 3. mα mod one 7 3.1. Geometry of Γ\G 9 3.2. Dynamics on Γ\G 10 3.3. Mixing and uniform distribution 12 4. √ mα mod one 14 4.1. The case α = 1 15 4.2. Some heuristics in the case α = √ 2 16 5. Ratner's theorem 19 5.1. Limit distributions of translates 19 5.2. Equidi...