scholarly journals Plünnecke inequalities for countable abelian groups

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Michael Björklund ◽  
Alexander Fish
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Lower Bounds ◽  
Correspondence Principle ◽  
Additive Basis ◽  
Banach Density ◽  
New Form ◽  
Countable Abelian Group ◽  
Product Sets ◽  
Measure Preserving Actions

AbstractWe establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers.We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

scholarly journals Plünnecke inequalities for measure graphs with applications

2015 ◽  
Vol 37 (2) ◽  
pp. 418-439
Author(s):  
KAMIL BULINSKI ◽  
ALEXANDER FISH
Keyword(s):  
Recent Work ◽  
Measure Space ◽  
Correspondence Principle ◽  
Abelian Groups ◽  
Density Estimates ◽  
Vertex Set ◽  
Banach Density ◽  
Measure Preserving Actions

We generalize Petridis’s new proof of Plünnecke’s graph inequality to graphs whose vertex set is a measure space. Consequently, by a recent work of Björklund and Fish, this gives new Plünnecke inequalities for measure-preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that extend those given by Jin.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

The number of sumsets in Abelian group

2020 ◽  
Vol 30 (5) ◽  
pp. 339-345
Author(s):  
Aleksandr A. Sapozhenko ◽  
Vahe G. Sargsyan
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Upper And Lower Bounds

AbstractAsymptotic upper and lower bounds for the numbers of distinct subsets A + B in Abelian group of order n are derived, where |A|, |B| ≥ n(log n)−1/8.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

An inverse theorem for additive bases

2016 ◽  
Vol 12 (06) ◽  
pp. 1509-1518 ◽  
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Inverse Theorem ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].

scholarly journals Mixing properties of induced random transformations

1997 ◽  
Vol 17 (4) ◽  
pp. 839-847 ◽  
Author(s):  
HANS-OTTO GEORGII
Keyword(s):  
Random Walk ◽  
Abelian Group ◽  
Probability Space ◽  
Return Time ◽  
Skew Product ◽  
Mixing Properties ◽  
Countable Abelian Group ◽  
Measure Preserving Transformations

Let $S(N)$ be a random walk on a countable abelian group $G$ which acts on a probability space $E$ by measure-preserving transformations $(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time $\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding induced skew product transformation is K-mixing whenever a natural subgroup of $G$ acts ergodically on $E$.

scholarly journals HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

2016 ◽  
Vol 81 (4) ◽  
pp. 1555-1562 ◽  
Author(s):  
MAURO DI NASSO ◽  
ISAAC GOLDBRING ◽  
RENLING JIN ◽  
STEVEN LETH ◽  
MARTINO LUPINI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Finite Subset ◽  
Amenable Group ◽  
High Density ◽  
Amenable Groups ◽  
Quantitative Version ◽  
Banach Density ◽  
Piecewise Syndetic ◽  
Product Sets

AbstractM. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd, this result was first proven by the current set of authors using completely different techniques.

Additive bases of Cp ⊕ Cpn

2017 ◽  
Vol 13 (09) ◽  
pp. 2453-2459
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group, and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. In this paper, we show that [Formula: see text] where [Formula: see text].

Multifractal dimensions of product measures

1996 ◽  
Vol 120 (4) ◽  
pp. 709-734 ◽  
Author(s):  
L. Olsen
Keyword(s):  
Probability Measure ◽  
Lower Bounds ◽  
Borel Probability Measure ◽  
Multifractal Spectrum ◽  
Upper And Lower Bounds ◽  
Packing Measure ◽  
Multifractal Spectra ◽  
Product Measures ◽  
Borel Probability ◽  
Image Position

AbstractWe study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

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