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Miklós Schweitzer 2003, Problem 4

Source: Miklós Schweitzer 2003

July 30, 2016
college contestsMiklos Schweitzer

Problem Statement

Let {an,1,…,an,n}n=1∞\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty} integers such that an,i≠an,ja_{n,i}\neq a_{n,j} for 1≤i<j≤n ,n=2,3,…1\le i<j\le n\, , n=2,3,\ldots and let ⟨y⟩∈[0,1)\left\langle y\right\rangle\in [0,1) denote the fractional part of the real number yy. Show that there exists a real sequence {xn}n=1∞\{ x_n\}_{n=1}^{\infty} such that the numbers ⟨an,1xn⟩,…,⟨an,nxn⟩\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle are asymptotically uniformly distributed on the interval [0,1][0,1].
(translated by L. Erdős)
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