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Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$.

Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is said to be uniformly distributed modulo 1 (abbreviated u.d. mod 1) if for every pair $a, b$ of real numbers with $0<a<b<1$ we have $$\lim\limits_{N\rightarrow\infty}\frac{A([a,b);N;\omega)}{N}=b-a.$$

Assume we have a real number sequence $(a_n):n\in\mathbb N$, which is u.d. mod 1. Intuition suggests that

$$P\left(\{a_n\}>\frac 1n\right)=1, \quad\text{when }\quad n\rightarrow\infty.$$

I am looking for a reference where this fact is proven and explored. Disproving my intuition would be welcome too.

Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$.

Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is said to be uniformly distributed modulo 1 (abbreviated u.d. mod 1) if for every pair $a, b$ of real numbers with $0<a<b<1$ we have $$\lim\limits_{N\rightarrow\infty}\frac{A([a,b);N;\omega)}{N}=b-a.$$

Assume we have a real number sequence $(a_n):n\in\mathbb N$, which is u.d. mod 1. It is obvious that for smaller intervals the proportion of fractional parts in those intervals will be smaller. Hence, intuition suggests that some kind of measure $P$ could be introduced to get

$$P\left(\{a_n\}>\frac 1n\right)=1, \quad\text{when }\quad n\rightarrow\infty.$$

I am looking for a reference where this idea is explored. Disproving my intuition for different kind of measures would be welcome too.

I do not want to reinvent the wheel and I am looking for a most common and conventional approach in building this bridge between number and probability theory.

Assume we have a real number sequence $(a_n):n\in\mathbb N$, which is u.d. mod 1. Intuition suggests that

$$P\left(\{a_n\}>\frac 1n\right)=1, \quad\text{when }\quad n\rightarrow\infty.$$

I am looking for a reference where this fact is proven and explored. Disproving my intuition would be welcome too.

Define counting function $A(E; N; \omega)$ as the number of terms $x_n, 1\leq n\leq N$, for which $\{x_n\}\in E$.

Then the sequence $\omega=(x_n), n=1,2,...,$ of real numbers is said to be uniformly distributed modulo 1 (abbreviated u.d. mod 1) if for every pair $a, b$ of real numbers with $0<a<b<1$ we have $$\lim\limits_{N\rightarrow\infty}\frac{A([a,b);N;\omega)}{N}=b-a.$$

Assume we have a real number sequence $(a_n):n\in\mathbb N$, which is u.d. mod 1. Intuition suggests that

$$P\left(\{a_n\}>\frac 1n\right)=1, \quad\text{when }\quad n\rightarrow\infty.$$

I am looking for a reference where this fact is proven and explored. Disproving my intuition would be welcome too.