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UNIFORM DISTRIBUTION AND ALGORITHMIC RANDOMNESS JEREMY AVIGAD
 

Summary: UNIFORM DISTRIBUTION AND ALGORITHMIC RANDOMNESS
JEREMY AVIGAD
Abstract. A seminal theorem due to Weyl [10] states that if (an) is any sequence
of distinct integers, then, for almost every x R, the sequence (an x) is uniformly
distributed modulo one. In particular, for almost every x in the unit interval,
the sequence (an x) is uniformly distributed modulo one for every computable
sequence (an) of distinct integers. I show that every Schnorr-random real has
this property, and raise questions as to how this property compares with other
notions of randomness.
1. Introduction
If x is a real number, let {x} denote the fractional part of x, and let denote
Lebesgue measure. A sequence (xn)nN of real numbers is said to be uniformly
distributed modulo one if for every interval I [0, 1],
lim
n
|{i < n | {xi} I}|
n
= (I).
In words, a sequence is uniformly distributed modulo one if the limiting frequency
with which it visits any given interval is what one would expect if the elements of

  

Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University

 

Collections: Multidisciplinary Databases and Resources; Mathematics