2,511 research outputs found
Minimum number of additive tuples in groups of prime order
For a prime number and a sequence of integers , let be the minimum number of
-tuples with
, over subsets of
sizes respectively. An elegant argument of Lev (independently
rediscovered by Samotij and Sudakov) shows that there exists an extremal
configuration with all sets being intervals of appropriate length, and
that the same conclusion also holds for the related problem, reposed by Bajnok,
when and , provided is not equal 1 modulo
. By applying basic Fourier analysis, we show for Bajnok's problem that if
and are fixed while tends to
infinity, then the extremal configuration alternates between at least two
affine non-equivalent sets.Comment: This version is the same as the published version except for
modifications to reflect Reference [5], that was brought to our attention
after publicatio
Enumeration of three term arithmetic progressions in fixed density sets
Additive combinatorics is built around the famous theorem by Szemer\'edi
which asserts existence of arithmetic progressions of any length among the
integers. There exist several different proofs of the theorem based on very
different techniques. Szemer\'edi's theorem is an existence statement, whereas
the ultimate goal in combinatorics is always to make enumeration statements. In
this article we develop new methods based on real algebraic geometry to obtain
several quantitative statements on the number of arithmetic progressions in
fixed density sets. We further discuss the possibility of a generalization of
Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3:
Incorporated feedbac
Nonuniform Fuchsian codes for noisy channels
We develop a new transmission scheme for additive white Gaussian noisy (AWGN)
channels based on Fuchsian groups from rational quaternion algebras. The
structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence
conventional decoding methods based on linearity and symmetry do not apply.
Previously, only brute force decoding methods with complexity that is linear in
the code size exist for general nonuniform codes. However, the properly
discontinuous character of the action of the Fuchsian groups on the complex
upper half-plane translates into decoding complexity that is logarithmic in the
code size via a recently introduced point reduction algorithm
Discrete Kakeya-type problems and small bases
A subset U of a group G is called k-universal if U contains a translate of
every k-element subset of G. We give several nearly optimal constructions of
small k-universal sets, and use them to resolve an old question of Erdos and
Newman on bases for sets of integers, and to obtain several extensions for
other groups.Comment: 12 page
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
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