2,069 research outputs found
van Douwen's problems related to the Bohr topology
We comment van Douwen's problems on the Bohr topology of the abelian groups
raised in his paper (The maximal totally bounded group topology on G and the
biggest minimal G-space for Abelian groups G) as well as the steps in the
solution of some of them. New solutions to two of the resolved problems are
also given.Comment: 14 page
Chowla's cosine problem
Suppose that G is a discrete abelian group and A is a finite symmetric subset
of G. We show two main results. i) Either there is a set H of O(log^c|A|)
subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X
on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then
either there is a subgroup H of G such that |A \triangle H| = o(|A|), or there
is a character X on G such that -wh{1_A}(X)>> |A|^c.Comment: 21 pp. Corrected typos. Minor revision
Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups
It is shown that distance powers of an integral Cayley graph over an abelian
group are again integral Cayley graphs over that group. Moreover, it is proved
that distance matrices of integral Cayley graphs over abelian groups have
integral spectrum
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
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