
Let N be a positive integer. A sequence X=(x_{1},x_{2},…,x_{N}) of points in the unit interval [0,1) is piercing if {x_{1},x_{2},…,x_{n}}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. 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. We study a more general variant of piercing sequences. Let f(n)≥n be an infinite nondecreasing sequence of positive integers. A sequence X=(x_{1},x_{2},…,x_{f(N)}) is fpiercing if {x_{1},x_{2},…,x_{f(n)}}∩[i/n,(i+1)/n)≠∅ holds for every n=1,2,…,N and every i=0,1,…,n−1. A special case of f(n)=n+d, with d a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each d≥0, the maximum length of any (n+d)piercing sequence is finite. Expressing this maximum length as s(d)+d, they obtained an exponential upper bound on the function s(d), which was later improved to s(d)=O(d^{3}) by Graham and Levy. Recently, Konyagin proved that 2d⩽s(d)<200d holds for all sufficiently big d. Using a different technique based on the Farey fractions and stickbreaking games, we prove here that the function s(d) satisfies ⌊c_{1}d⌋⩽s(d)⩽c_{2}d+o(d), where c_{1}=ln2/(1−ln2)≈2.25 and c_{2}=(1+ln2)/(1−ln2)≈5.52. We also prove that there exists an infinite fpiercing sequence with f(n)=γn+o(n) if and only if γ≥1/ln2≈1.44. This is joint work with Marcin Anholcer, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Rafał Pyzik, and Mariusz Zając.
 Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Rafał Pyzik, Mariusz Zając. On a Problem of Steinhaus. arXiv:2111.01887. (2021).
