On a problem of Steinhaus

Abstract

AbstractLet be a positive integer. A sequence of points in the unit interval [0,1) is piercing if holds for every and every . In 1958, Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most 74 elements. This was later improved to the best possible value of 17 by Warmus, and independently by Berlekamp and Graham. In this paper, we study a more general variant of piercing sequences. Let be an infinite nondecreasing sequence of positive integers. A sequence is ‐piercing if holds for every and every . A special case of , with a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each , the maximum length of any ‐piercing sequence is finite. Expressing this maximum length as , they obtained an exponential upper bound on the function , which was later improved to by Graham and Levy. Recently, Konyagin proved that holds for all sufficiently big . Using a different technique based on the Farey fractions and stick‐breaking games, we prove here that the function satisfies where and . We also prove that there exists an infinite ‐piercing sequence with if and only if .