Nine-distance theorem in three-dimensional space

Abstract: The classical three-distance theorem (also known as the Steinhaus conjecture) states that the gaps determined by a finite one-dimensional Kronecker sequence have at most three distinct lengths. We prove that its natural $d$-dimensional analogue has at most $2^d+1$ distinct lengths, which is the optimal bound in dimensions $d=1,2,3$. This proves the conjecture of Haynes and Marklof about 3-dimensional distances.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

---

One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

For questions or subscribing to the mailing list, contact the organisers at numeration@irif.fr

---