scholarly journals An Elementary Proof of Jin's Theorem with a Bound

10.37236/3846 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
Difference Sets ◽  
Explicit Bound ◽  
Finite Set

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If $A$ and $B$ are sets of integers having positive upper Banach densities $a$ and $b$ respectively, then there exists a finite set $F$ of cardinality at most $1/ab$ such that $(A-B)+F$ covers arbitrarily long intervals.

Another proof of a theorem on difference sets

1967 ◽  
Vol 63 (3) ◽  
pp. 595-596
Author(s):  
D. L. Yates
Keyword(s):  
Cyclic Group ◽  
Elementary Proof ◽  
Existence Theorems ◽  
Difference Sets ◽  
Difference Set ◽  
En Bloc ◽  
Elementary Matrix

Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

scholarly journals “Statistics 103” for Multitarget Tracking

Sensors ◽  
2019 ◽  
Vol 19 (1) ◽  
pp. 202 ◽  
Author(s):  
Ronald Mahler
Keyword(s):  
Point Process ◽  
Information Fusion ◽  
Theoretical Foundation ◽  
Measure Theory ◽  
Multitarget Tracking ◽  
Mathematical Abstraction ◽  
Finite Set ◽  
Theoretic Point

The finite-set statistics (FISST) foundational approach to multitarget tracking and information fusion was introduced in the mid-1990s and extended in 2001. FISST was devised to be as “engineering-friendly” as possible by avoiding avoidable mathematical abstraction and complexity—and, especially, by avoiding measure theory and measure-theoretic point process (p.p.) theory. Recently, however, an allegedly more general theoretical foundation for multitarget tracking has been proposed. In it, the constituent components of FISST have been systematically replaced by mathematically more complicated concepts—and, especially, by the very measure theory and measure-theoretic p.p.’s that FISST eschews. It is shown that this proposed alternative is actually a mathematical paraphrase of part of FISST that does not correctly address the technical idiosyncrasies of the multitarget tracking application.

scholarly journals On minimal self-joinings in topological dynamics

1987 ◽  
Vol 7 (2) ◽  
pp. 211-227 ◽  
Author(s):  
Andrés del Junco
Keyword(s):  
Automorphism Group ◽  
Ergodic Theory ◽  
Metric Space ◽  
Topological Dynamics ◽  
Orbit Closure ◽  
Compact Metric Space ◽  
Minimal Self ◽  
Finite Set ◽  
Countably Infinite

AbstractIf X is a compact metric space and T a homeomorphism of X we say (X, T) has almost minimal power joinings (AMPJ) if there is a dense GδX* in X such that for each finite set k, x∈(X*)k and l:k → ℤ−{0}, the orbit closure cl {} is a product of off-diagonals (POOD) on Xk. By an offdiagonal on Xk′, k′k we mean a set of the form (⊗,j∈k′Tm(j))Δ, Δ the diagonal in Xk′, m:k′→ℤ any function, and by a POOD on Xk we mean that k is split into subsets k′, on each Xk′ we put an off-diagonal and then we take the product of these.We show that examples of AMPJ exist and that this definition leads to a theory completely analogous to Rudolph's theory of minimal self-joinings in ergodic theory. In particular if (X, T) has AMPJ the automorphism group of T is {Tn}, T has only almost 1-1 factors (other than the trivial one) and the automorphism group and factors of ⊕i ∊ kT, k finite or countably infinite, can be very explicitly described. We also discuss ℝ-actions.

On the number of holomorphic mappings between Riemann surfaces of finite analytic type

2011 ◽  
Vol 54 (3) ◽  
pp. 711-730
Author(s):  
Yoichi Imayoshi ◽  
Manabu Ito ◽  
Hiroshi Yamamoto
Keyword(s):  
Riemann Surfaces ◽  
Holomorphic Mappings ◽  
Explicit Bound ◽  
Finiteness Result ◽  
Classical Statement ◽  
Finite Set ◽  
Compact Riemann Surfaces ◽  
Set Of Points ◽  
Topological Data

AbstractThe set of non-constant holomorphic mappings between two given compact Riemann surfaces of genus greater than 1 is always finite. This classical statement was made by de Franchis. Furthermore, bounds on the cardinality of the set depending only on the genera of the surfaces have been obtained by a number of mathematicians. The analysis is carried over in this paper to the case of Riemann surfaces of finite analytic type (i.e. compact Riemann surfaces minus a finite set of points) so that the finiteness result, together with a crude but explicit bound depending only on the topological data, may be extended for the number of holomorphic mappings between such surfaces.

scholarly journals Vladimir Abramovich Rokhlin (23 August 1919 to 3 December 1984): Introductory note

1989 ◽  
Vol 9 (4) ◽  
pp. 605-608
Author(s):  
A. Katok
Keyword(s):  
Ergodic Theory ◽  
Measure Theory ◽  
The Other ◽  
Natural Evolution ◽  
And Topology ◽  
Introductory Note ◽  
Time Periods ◽  
The Right ◽  
Short Time ◽  
The Impact

The Ergodic Theory and Dynamical Systems journal pays tribute to V. A. Rokhlin, one of the founders of ergodic theory, a world-renowned topologist and geometer, and a man of tragic fate and exceptional courage. Rokhlin's mathematical heritage splits rather sharply into the ergodic theory – measure theory and topology – geometry parts. This fact has to do with a natural evolution of his interests but also with the keen sense of style in mathematics that Rokhlin possessed to an unusual degree. Naturally, we will concentrate on Rokhlin's contributions to ergodic theory and measure theory, his influence on other mathematicians working in those fields, and the development of some of his ideas. Fortunately, the topology part of Rokhlin's heritage has been superbly presented in Part I of the book ‘A la recherche de la topologie perdue’ published by Birkhaüser in Progress in Mathematics series (v. 62, 1986). The same cannot be said about the work on real algebraic geometry, Rokhlin's last big achievement. The impact of that work, carried out by his students, however, is very much felt now, and is very unlikely to be forgotten or neglected. On the other hand, his work in ergodic theory and measure theory, was restricted primarily to two relatively short time periods, 1947–1950 and 1959–1964, and for the most part was not followed by Rokhlin's immediate students. Hence, it runs a certain risk of being underestimated. We hope that the articles by A. Vershik; S. Yuzvinsky and B: Weiss, published in this issue, will put Rokhlin's work and his influence in ergodic theory into the right perspective.

scholarly journals NORMAL NUMBERS AND COMPLETENESS RESULTS FOR DIFFERENCE SETS

2017 ◽  
Vol 82 (1) ◽  
pp. 247-257 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Ergodic Theory ◽  
Difference Sets ◽  
Normal Numbers ◽  
Real Numbers ◽  
Image Position

AbstractWe consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes ${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$ and ${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively, ω-differences) of ${\bf{\Pi }}_3^0 $ sets.

scholarly journals Sumset and Inverse Sumset Theory for Shannon Entropy

2010 ◽  
Vol 19 (4) ◽  
pp. 603-639 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Shannon Entropy ◽  
Difference Sets ◽  
Additive Group ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Finite Sets ◽  
Finite Set ◽  
Torsion Free ◽  
Sharp Lower Bound ◽  
Bounded Rank

Let G = (G, +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets A − B, while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set A ⊂ G is replaced by a discrete random variable X taking values in G, and the cardinality |A| is replaced by the Shannon entropy H(X). In particular, we classify those random variables X which have small doubling in the sense that H(X1 + X2) = H(X) + O(1) when X1, X2 are independent copies of X, by showing that they factorize as X = U + Z, where U is uniformly distributed on a coset progression of bounded rank, and H(Z) = O(1).When G is torsion-free, we also establish the sharp lower bound $\Ent(X+X) \geq \Ent(X) + \frac{1}{2} \log 2 - o(1)$, where o(1) goes to zero as H(X) → ∞.

A short proof of the Ornstein theorem

2011 ◽  
Vol 32 (2) ◽  
pp. 587-597
Author(s):  
T. DOWNAROWICZ ◽  
J. SERAFIN
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Short Proof ◽  
Bernoulli Shifts ◽  
The Core

AbstractWe give a short (and fairly elementary) proof of the ‘residual Sinai theorem’ of which the Ornstein theorem is an immediate consequence. In principle, we follow the lines of [BKS] but our proof avoids any substantial quotations, in particular, invoking any characterizations of systems isomorphic to Bernoulli shifts. The core of the proof is five pages long and it relies only on standard facts in ergodic theory.

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