Tiling sets and spectral sets over finite fields
| Autor: | H. Liu, C. Rojas Mena, J. Weill, E. Bau, C. Yu, Adam Lott, B. Ayachi, Giorgis Petridis, S. Nan, C. Aten, A. Sheikh, T. Tribone, Jonathan Pakianathan, Alex Iosevich, I. MacKinnon, S. Maimon, D. FitzPatrick |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Conjecture
Mathematics - Number Theory Mathematics::Number Theory 010102 general mathematics 01 natural sciences Spectral set Prime (order theory) 010101 applied mathematics Combinatorics 46S10 52C22 05D99 Finite field Mathematics - Classical Analysis and ODEs Classical Analysis and ODEs (math.CA) FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) Number Theory (math.NT) Locally compact space 0101 mathematics Abelian group Analysis Vector space Counterexample Mathematics |
| Popis: | We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6] ). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z p 5 for all odd primes p and Z p 4 for all odd primes p such that p ≡ 3 mod 4 . Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|