The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials

By I. Łaba. 2002

61 citations recorded by the source corpus.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Some reductions of the spectral set conjecture to integers (2013, background): “(It was proved in [9] that Ł is a spectrum for A and, with Proposition 2·5, Ł is a spectrum for .”
Fuglede's conjecture is false in 5 and higher dimensions (2003, result): “For instance, the results of [ 13 ] show that Fuglede’s conjecture is true for cyclic p-groups ZpN, so one cannot directly replicate the above counterexample in one dimension.”
On Fuglede's Conjecture and the Existence of Universal Spectra (2006, result): “of the conjecture, when the spectrum or the tiling complement is assumed to be a lattice. Later on, for many years several positive results seemed to indicate the validity of the conjecture (see e.g. [5, 4, 6, 9, 10, 11, 13, 15]. Recently, however, Tao [17] disproved the “spectral → tile” direction ”
On Spectral Cantor Measures (2001, background): “The argument below is essentially a repetition of the proof of Theorem 1.5(i) in [ 13 ] specialized to the case of two prime factors.”
Fuglede’s conjecture fails in dimension 4 (2005, result): “was recently proved in 2 dimensions for convex bodies in [5]. Even for ﬁnite union of intervals the conjecture is still open in dimesion 1. Some partial results concerning this case appeared in [10], [11]. In arbitirary dimensions certain classes of nontiling sets are already known to admit no spect”