Uniform equipartition test bounds for multiply sequences (Q2897025)

The paper deals with the sequence \(x_n=ax_{n-1}\pmod 1\), with \(a\) being a non-negative integer. Let NEWLINE\[NEWLINE P(\langle S_n\rangle)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum\limits_{S_n\text{is true}\atop 1\leq n\leq N}1 NEWLINE\]NEWLINE where \(S_n\) is a statement, then a sequence \(\langle x_n\rangle\) is \(m\)-equipartitioned if for any permutation \(i_1,\ldots,i_m\) of the set \(\{i,\ldots, i+m-1\}\) one has NEWLINE\[NEWLINEP(x_{i_1}>\ldots>x_{i_m})=\frac{1}{m!}.NEWLINE\]NEWLINE By exact calculation of different \(P(x_{i_1}>\ldots>x_{i_m})\) it is proved that the sequence \(x_n=ax_{n-1}\pmod{1}\) is not \(m\)-equipartitioned for \(m>2\).