
This paper presents a new construction of the
Let
We prove that a sumset of a TE subset of N (these sets can be viewed as “aperiodic” sets) with a set of positive upper density intersects any polynomial sequence. For WM sets (subclass of TE sets) we prove that the intersection has lower Banach density one. In addition we obtain a generalization of the latter result to the case of several polynomials.
In this paper, we prove the Tiling implies Spectral part of Fuglede’s cojecture for the three interval case. Then we prove the converse Spectral implies Tiling in the case of three equal intervals and also in the case where the intervals have lengths 1/2, 1/4, 1/4. Next, we consider a set Ω ⊂ R, which is a union of n intervals. If Ω is a spectral set, we prove a structure theorem for the spectrum provided the spectrum is assumed to be contained in some lattice. The method of this proof has some implications on the Spectral implies Tiling part of Fuglede’s conjecture for three intervals. In the final step in the proof, we need a symbolic computation using Mathematica. Finally with one additional assumption we can conclude that the Spectral implies Tiling holds in this case.
We show that if
We provide further explanation of the significance of an example in a recent paper of Wolf in the context of the problem of finding large subspaces in sumsets.
Lucy Slater used Bailey’s
In the present paper, we apply the same techniques to Chu’s
In re-examining Slater’s work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the
Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the
In [Fr2, Skr], Frolov and Skriganov showed that low discrepancy point sets in the multidimensional unit cube
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