Digit patterns in real numbers created from permutations

Abstract. Given a positive integer \(k\), we construct a binary number \(0.a_1a_2a_3\ldots\) having the property that any sequence \(a_{m+1}\ldots a_{m+k}\) of \(k\) consecutive digits from its binary expansion appears with a frequency directly related to the various permutations of the set \(\{1,2,\ldots,k+1\}\).