601 research outputs found
New examples of complete sets, with connections to a Diophantine theorem of Furstenberg
No full textA set is called if every sufficiently large integer can be written as the sum of distinct elements of . In this paper we present a new method for proving the completeness of a set, improving results of Cassels ('60), Zannier ('92), Burr, Erd\H{o}s, Graham, and Li ('96), and Hegyv\'ari ('00). We also introduce the somewhat philosophically related notion of a set and refine a theorem of Furstenberg ('67)
Improving dimension estimates for Furstenberg-type sets
No full textIn this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH(ℓe∩F)⩾α. It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets.
The main difficulty we had to overcome, was that if Hh(F)=0, there always exists g≺h such that Hg(F)=0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α=0.Agencia Nacional de Promoción Científica y Tecnológica (Argentina)Universidad de Buenos Aire
Projections, Furstenberg sets, and the sum-product problem
No full textWe make progress on several interrelated problems at the intersection of
geometric measure theory, additive combinatorics and harmonic analysis: the
discretised sum-product problem, exceptional estimates for orthogonal
projections, and the dimension of Furstenberg sets.
We give a new proof of the following asymmetric sum-product theorem: Let
be Borel sets with . Then, there exists such that
Here we only mention
special cases of our results on projections and Furstenberg sets. We prove that
every -Furstenberg set has Hausdorff dimension We prove
that every -Furstenberg set associated with a
-Ahlfors-regular line set has Let denote projection onto
the line spanned by . We prove that if is a Borel set with , then whenever , and the factor "" on the right-hand side can be
omitted if is Ahlfors-regular.Comment: 73 pages. v4: improved Theorem 5.61 and Remark 5.6
Slices and distances: on two problems of Furstenberg and Falconer
No full textWe survey the history and recent developments around two decades-old problems that continue to attract a great deal of interest: the slicing , conjecture of H. Furstenberg in ergodic theory, and the distance set problem in geometric measure theory introduced by K. Falconer. We discuss some of the ideas behind our solution of Furstenberg\u27s slicing conjecture, and recent progress in Falconer\u27s problem. While these two problems are on the surface rather different, we emphasize some common themes in our approach: analyzing fractals through a combinatorial description in terms of ``branching numbers\u27\u27, and viewing the problems through a ``multiscale projection\u27\u27 lens.25 pages, submitted to the Proceedings ofthe ICM 202
Furstenberg boundaries for pairs of groups
No full textFurstenberg has associated to every topological group G a universal boundary partial derivative(G). If we consider in addition a subgroup H < G, the relative notion of (G,H)-oundaries admits again a maximal object partial derivative(G, H). In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex Delta(G, H), namely the simplex of measures on partial derivative(G, H). We determine the boundary partial derivative(G, H) in a number of cases, highlighting properties that might appear unexpected.EGGCI
Dimensions of Furstenberg sets and an extension of Bourgain\u27s projection theorem
No full textWe show that the Hausdorff dimension of -Furstenberg sets is at least , where depends only on and . This improves the previously best known bound for , in particular providing the first improvement since 1999 to the dimension of classical -Furstenberg sets for . We deduce this from a corresponding discretized incidence bound under minimal non-concentration assumptions, that simultaneously extends Bourgain\u27s discretized projection and sum-product theorems. The proofs are based on a recent discretized incidence bound of T.~Orponen and the first author and a certain duality between and -Furstenberg sets.15 page
A Furstenberg Transformation of the 2-Torus Without Quasi-Discrete Spectrum
No full textAbstractR. Ji asked whether or not a Furstenberg transformation of the 2-torus of the form (x,y) → (e2πiθx, f(x)y), where θ is irrational and f : T —> T is continuous with non-zero degree k, is topologically conjugate to the Anzai transformation (x, y) → (e2πiθx, xk y) or its inverse. In this paper this question is settled in the negative. Further, some sufficient conditions are given under which the crossed product C*-algebra associated with a Furstenberg transformation of the 2-torus has a unique tracial state.</jats:p
Furstenberg measure and Iterated Function Systems with inverses (Integrated Research on Random Dynamical Systems and Multi-Valued Dynamical Systems)
No full textMotivated by the study of the Furstenberg measure, in [1] the author introduced Iterated Function Systems with inverses (i.e. IFS that contain inverse maps). In this note we present a conjecture
Integrability of orthogonal projections, and applications to Furstenberg sets
No full textLet be the Grassmannian manifold of -dimensional
subspaces of , and let be
the orthogonal projection. We prove that if is a compactly supported
Radon measure on satisfying the -dimensional Frostman
condition for all and ,
then The
upper bound for is sharp, at least, for , and every .
Our motivation for this question comes from finding improved lower bounds on
the Hausdorff dimension of -Furstenberg sets. For and
, a set is called an
-Furstenberg set if there exists a -dimensional family
of affine lines in such that for all . As a consequence of our projection
theorem in , we show that every -Furstenberg set with satisfies This improves on previous bounds for pairs
with and for a small absolute constant
. We also prove a higher dimensional analogue of this estimate
for codimension-1 Furstenberg sets in . As another corollary of
our method, we obtain a -discretised sum-product estimate for
-sets. Our bound improves on a previous estimate of Chen for every
, and also of Guth-Katz-Zahl for .Comment: 28 pages, 3 figures. v3: reviewer comments incorporated, to appear in
Adv. Mat
Conditional Indexation Bias in Yields Reported on Inflation-Indexed Securities with Special Reference to UDIBONOS and TIPS
No full textThe real rate of return on inflation-indexed government securities is calculated and published as if indexation succeeded perfectly in keeping the real value of coupon and principal payments unchanged. In fact the procedure of indexing to the lagged momentum of the seasonally unadjusted CPI gives rise to three types of indexation bias that may change the expected real value of the future stream of payments in relation to the current par value. These biases are due to i) seasonality, ii) non-seasonal fluctuations in reported inflation rates, and iii) any expected “permanent” changes in future rates of inflation (or the reporting thereof) being capable of creating predictable changes in the real value of the inflation-adjusted principal with the indexation procedure actually in force. They are one more, directly quantifiable, reason why the reported yields do not provide the long-sought definite revelation of the riskless real rate of interest and hence of the expected rate of inflation by comparison with nominal interest rates.
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