uniformly fractional parts

Given $A,B$ u.d. in [0,1] and $k\in\mathbb{N}$ we define $X:=A+kB-\{A+kB\}$. How to prove $X$ is u.d..

I read something about the "Weyl equidistribution theorem" (by google). But I never heard sth about ergodic theory before. I wonder if there is literature or an easy proof for the statement above. I try to show it directly with cdf. This statement is a generalization of an exercise in "Morgan, B.J.T., Elements of Simulation" (p.72).

$\{\bullet\}$ means the largest integer

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1 Answer

Here is a general result, which implies the one you want (try $C=kB$, for any $k$):

For every independent $A$ and $C$ such that $A$ is uniform on $[0,1]$, $X=A+C-\{A+C\}$ is uniform on $[0,1]$.

To prove this, note that, conditionally on $[C=c]$, $A+C$ is uniform on the interval $[n(c)+x(c),n(c)+x(c)+1]$, where $n(c)$ is the integer part of $c$ and $x(c)$ is its fractional part. The subinterval $[n(c)+x(c),n(c)+1]$ yields $X$ uniform on $[x(c),1]$, with probability $1-x(c)$, while the subinterval $[n(c)+1,n(c)+x(c)+1]$ yields $X$ uniform on $[0,x(c)]$, with probability $x(c)$. Summing these two contributions, one sees that, conditionally on $[C=c]$, $X$ is uniform on $[0,1]$. This holds for every $c$ hence $X$ is uniform on $[0,1]$, unconditionally.

Note that the hypothesis about the distribution of $B$ is not needed.

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