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Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
A multi-dimensional Szemer\'edi theorem for the primes via a correspondence principle
We establish a version of the Furstenberg-Katznelson multi-dimensional
Szemer\'edi in the primes , which roughly
speaking asserts that any dense subset of contains
constellations of any given shape. Our arguments are based on a weighted
version of the Furstenberg correspondence principle, relative to a weight which
obeys an infinite number of pseudorandomness (or "linear forms") conditions,
combined with the main results of a series of papers by Green and the authors
which establish such an infinite number of pseudorandomness conditions for a
weight associated with the primes. The same result, by a rather different
method, has been simultaneously established by Cook, Magyar, and Titichetrakun.Comment: 20 pages, no figures. Submitted, Israel J. Math. Several suggestions
of an anonymous referee have been implemente
On the computation of rational points of a hypersurface over a finite field
We design and analyze an algorithm for computing rational points of
hypersurfaces defined over a finite field based on searches on "vertical
strips", namely searches on parallel lines in a given direction. Our results
show that, on average, less than two searches suffice to obtain a rational
point. We also analyze the probability distribution of outputs, using the
notion of Shannon entropy, and prove that the algorithm is somewhat close to
any "ideal" equidistributed algorithm.Comment: 31 pages, 5 table
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