Notes on uniform distribution modulo one

On uniform distribution modulo one

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

Uniform Distribution modulo One of Some Sequences concerning the Euler Function

In this paper, we follow the recent method in the theory of uniform distribution, developed by J.-M. Deshouillers and H. Iwaniec, to prove uniform distribution modulo one of various sequences involving the Euler function, together with some generalizations.

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...

The distribution of α p modulo one 269

We prove that, for any irrational number α, there are infinitely many primes p such that ‖αp‖ < p−1/3+ . Here ‖y‖ denotes the distance from y to the nearest integer. The proof uses Harman’s sieve method with arithmetical information coming from bounds for averages of Kloosterman sums.

On the Distribution of Inverses Modulo n