# Andras Biro: Tilings of the integers

If and every integer can be written uniquely as with and , then we write , and we say that is a tiling of by . If and is finite, it follows from the pigeon-hole principle that is periodic, i.e. with some positive integer . Let . If the diameter of is , and the least period of is , the pigeon-hole principle gives that . This result was recently improved by I.Z. Ruzsa, who proved in [3] that . A slightly weaker result was proved independently by M. Kolountzakis in [2]. In the other direction, the best known result is that is not true, see [2]. We see that these upper and lower estimates are very far from each other. In the present talk we sketch the proof of the new upper bound proved in [1]:

However, the problem whether a polynomial upper bound can be given for or not, remains open. The new upper bound will be a corollary of a theorem on the divisibility of integer polynomials.

## Bibliography

1
András Biró.
Divisibility of integer polynomials and tilings of the integers.
(submitted to Acta Arithmetica).

2
Mihail N. Kolountzakis.
Translational tilings of the integers with long periods.
Electron. J. Combin., 10:Research Paper 22, 9 pp. (electronic), 2003.

3
Robert Tijdeman (appendix by Imre Z. Ruzsa).
Periodicity and almost periodicity.
(preprint available at http://www.math.leidenuniv.nl/ tijdeman/preprints.html), 2002.