A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

Sam Mattheus

A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

Combinatorics 2019-04-30 v2 Classical Analysis and ODEs

Authors: Sam Mattheus

Abstract

In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$ , $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$ . On the contrary we show that the conjecture does hold in $(\mathbb{Z}/2\mathbb{Z})^4$ .

Keywords

fuglede conjecture spectral set non-tiling set finite abelian group counterexample

Cite

@article{arxiv.1904.11537,
  title  = {A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes},
  author = {Sam Mattheus},
  journal= {arXiv preprint arXiv:1904.11537},
  year   = {2019}
}

Comments

This result was found simultaneously and independently with [arXiv:1901.08734]. However, it contained an error which was fixed afterwards (again independently). We record it here as it gives a more geometrical approach to more or less the same construction

A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

Abstract

Keywords

Cite

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