Irregularity of distributions

Hugo Steinhaus asked a question about the possibility of regular distributions of fractions. His question was whether for every positive integer "N" there is a sequence of real numbers

:$x\_1,ldots,x\_N$

such that for every $scriptstyle\; n,in,\{1,,ldots,,N\}$ and every $scriptstyle\; k,in,\{1,,dots,,n\}$ we have

:

for some $scriptstyle\; i,in,\{1,,dots,,n\}$.

The answer that for "N" ≥ 18 it is not possible. It is possible for "N" ≤ 17; one example is

So, for example, with "n" = 5, we have 0 ≤ "x"₄ ≤ 1/5 ≤ "x"₂ ≤ 2/5 ≤ "x"₁ ≤ 3/5 ≤ "x"₅ ≤ 4/5 ≤ "x"₃ ≤ 1.

References

* H. Steinhaus, "One hundred problems in elementary mathematics", Basic Books, New York, 1964
* M. Warmus, "A Supplementary Note on the Irregularities of Distributions", "Journal of Number Theory" 8, 260–263, 1976.