23,630 research outputs found
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
A new approach to the periodicity lemma on strings with holes
We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a "wild card", a "don't-care" or an "indeterminate letter" in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [J. Berstel, L. Boasson, Partial words and a theorem of Fine and Wilf, Theoret. Comput. Sci. 218 (1999) 135-141]. We then study the two-hole case, where our result agrees with the one given in [F. Blanchet-Sadri, Robert A. Hegstrom, Partial words and a theorem of Fine and Wilf revisited, Theoret. Comput. Sci. 270 (1-2) (2002) 401-419] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally, we extend our result to three or more holes using elementary methods, and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum-length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from that used by Blanchet-Sadri and Hegstrom, and also simpler
Decidability of the HD0L ultimate periodicity problem
In this paper we prove the decidability of the HD0L ultimate periodicity
problem
A New Approach to the Periodicity Lemma on Strings with Holes
Abstract We first give an elementary proof of the periodicity lemma for strings containing one hole (variously called a "wild card" or a "don't-care" or an "indeterminate letter" in the literature). The proof is modelled on Euclid's algorithm for the greatest common divisor and is simpler than the original proof given in [BB99]. We then study the two hole case, where our result agrees with the one given in [BSH02] but is more easily proved and enables us to identify a maximum-length prefix or suffix of the string to which the periodicity lemma does apply. Finally we extend our result to three or more holes using elementary methods and state a version of the periodicity lemma that applies to all strings with or without holes. We describe an algorithm that, given the locations of the holes in a string, computes maximum length substrings to which the periodicity lemma applies, in time proportional to the number of holes. Our approach is quite different from the one in [BSH02, BS04] and also simpler
Simplicity of C*-algebras associated to row-finite locally convex higher-rank graphs
In previous work, the authors showed that the C*-algebra C*(\Lambda) of a
row-finite higher-rank graph \Lambda with no sources is simple if and only if
\Lambda is both cofinal and aperiodic. In this paper, we generalise this result
to row-finite higher-rank graphs which are locally convex (but may contain
sources). Our main tool is Farthing's "removing sources" construction which
embeds a row-finite locally convex higher-rank graph in a row-finite
higher-rank graph with no sources in such a way that the associated C*-algebras
are Morita equivalent.Comment: 18 pages, 1 figure, figure drawn using Tikz/PGF. Version 2: the
hypothesis "with no sources" has been removed from Theorem 3.4; it appeared
there in error since the main point of the theorem is that it applies in the
absence of this hypothesis (cf Theorem 3.1 of arXiv:math/0602120
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
Complete periodicity of Prym eigenforms
This paper deals with Prym eigenforms which are introduced previously by
McMullen. We prove several results on the directional flow on those surfaces,
related to complete periodicity (introduced by Calta). More precisely we show
that any homological direction is algebraically periodic, and any direction of
a regular closed geodesic is a completely periodic direction. As a consequence
we draw that the limit set of the Veech group of every Prym eigenform in some
Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at
infinity. We also construct new examples of translation surfaces satisfying the
topological Veech dichotomy. As a corollary we obtain new translation surfaces
whose Veech group is infinitely generated and of the first kind.Comment: 35 page
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