Irregularity of distributions explained

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

(k -1)/(n) \leq x _{i} < (k)/(n) .

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{align} x _{1} & = 0 . 029 \\ x _{2} & = 0 . 971 \\ x _{3} & = 0 . 423 \\ x _{4} & = 0 . 71 \\ x _{5} & = 0 . 27 \\ x _{6} & = 0 . 542 \\ x _{7} & = 0 . 852 \\ x _{8} & = 0 . 172 \\ x _{9} & = 0 . 62 \\ x _{10} & = 0 . 355 \\ x _{11} & = 0 . 777 \\ x _{12} & = 0 . 1 \\ x _{13} & = 0 . 485 \\ x _{14} & = 0 . 905 \\ x _{15} & = 0 . 218 \\ x _{16} & = 0 . 667 \\ x _{17} & = 0 . 324 \end{align}

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

0 < x _{1} < (1)/(5) < x _{5} < (2)/(5) < x _{3} < (3)/(5) < x _{4} < (4)/(5) < 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
• . . Irregularities in the distributions of finite sequences . Journal of Number Theory. 2. 1970. 2 . 152–161. 0269605. 10.1016/0022-314X(70)90015-6. 1970JNT.....2..152B . free.
• M. Warmus, "A Supplementary Note on the Irregularities of Distributions", Journal of Number Theory 8, 260 - 263, 1976.