Irregularity Of Distributions

The irregularity of distributions problem, stated first by Hugo Steinhaus, is a numerical problem with a surprising result. The problem is to find ''N'' numbers, $x_1,\ldots,x_N$, all between 0 and 1, for which the following conditions hold:

* The first two numbers must be in different halves (one less than 1/2, one greater than 1/2).
* The first 3 numbers must be in different thirds (one less than 1/3, one between 1/3 and 2/3, one greater than 2/3).
* The first 4 numbers must be in different fourths.
* The first 5 numbers must be in different fifths.
* etc.

Mathematically, we are looking for a sequence of real numbers

:$x_1,\ldots,x_N$

such that for every ''n'' ∈  and every ''k'' ∈  there is some ''i'' ∈  such that

:$\frac \leq x_i < \frac.$

Solution

The surprising result is that there is a solution up to ''N'' = 17, but starting at ''N'' = 18 and above it is impossible. A possible solution for ''N'' ≤ 17 is shown diagrammatically on the right; numerically it is as follows:

: $\begin x_ & = 0.029 \\ x_ & = 0.971 \\ x_ & = 0.423 \\ x_ & = 0.71 \\ x_ & = 0.27 \\ x_ & = 0.542 \\ x_ & = 0.852 \\ x_ & = 0.172 \\ x_ & = 0.62 \\ x_ & = 0.355 \\ x_ & = 0.777 \\ x_ & = 0.1 \\ x_ & = 0.485 \\ x_ & = 0.905 \\ x_ & = 0.218 \\ x_ & = 0.667 \\ x_ & = 0.324 \end$

In this example, considering for instance the first 5 numbers, we have

: $0 < x_1 < \frac < x_5 < \frac < x_3 < \frac < x_4 < \frac < x_2 < 1.$

Mieczysław Warmus concluded that 768 (1536, counting symmetric solutions separately) distinct sets of intervals satisfy the conditions for ''N'' = 17.

References

* H. Steinhaus, ''One hundred problems in elementary mathematics'', Basic Books, New York, 1964, page 12
*
* M. Warmus, "A Supplementary Note on the Irregularities of Distributions", '' Journal of Number Theory'' 8, 260–263, 1976.

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Fractions (mathematics)