Franklin's Notes

Uniform distribution modulo 1

Let $\omega=(x_n)$ be a sequence of real numbers, and define a counting function $\mathrm{A}(E;N;\omega)$ to denote the number of terms $x_n$ with $1\leq n\leq N$ such that the fractional part ${x_n}\in E$. Then the sequence $\omega$ is said to be uniformly distributed modulo 1 if and only if for all half-open intervals $[a,b)$ with $0\leq a<b\leq 1$.

Proposition 1. (Kuipers and Niederreiter, Exercise 1.1) A sufficient condition for $\omega=(x_n)$ to be uniformly distributed mod $1$ is for all $0\leq c\leq 1$.

Proof. If the given condition holds, then we have that for any $0\leq a < b\leq 1$, and so we may subtract these limits to obtain or which makes $\omega$ uniformly distributed modulo $1$. $\blacksquare$

Proposition 2. (Kuipers and Niederreiter, Exercise 1.4) A sufficient condition for $\omega=(x_n)$ to be uniformly distributed mod $1$ is that for all $a,b\in\mathbb{Q}$ with $0\leq a<b\leq 1$.

Proposition 3. (Kuipers and Niederreiter, Exercise 1.7-1.8) If finitely many terms are omitted or replaced in a sequence that is uniformly distributed modulo $1$, then the resulting sequence is also uniformly distributed modulo $1$.

Proposition 4. (Kuipers and Niederreiter, Theorem 1.1) A sequence $(x_n)$ is uniformly distributed mod $1$ if and only if, for every continuous function $f:[0,1]\to\mathbb{R}$, we have that

Proof. If $\chi_{[a,b)}$ denotes the characteristic function of the interval $[a,b)$, then the definition of uniform distribution is equivalent to the statement that for all $0\leq a<b\leq 1$. The desired result follows from the fact that continuous functions can be approximated to within arbitrary $L^1$ precision by sums of characteristic functions, and conversely characteristic functions can be approximated to within arbitrary $L^1$ precision by continuous functions. $\blacksquare$

Proposition 5. (Kuipers and Niederreiter, Corollary 1.2) A sequence $(x_n)$ is uniformly distributed mod $1$ if and only if, for every continuous function $f:\mathbb{R}\to\mathbb{R}$ with period $1$, we have that