PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

On sums of nearly affine Cantor sets

Volume 240 / 2018

Anton Gorodetski, Scott Northrup Fundamenta Mathematicae 240 (2018), 205-219 MSC: Primary 28A80, 37D99, 28A78. DOI: 10.4064/fm183-3-2017 Published online: 7 August 2017

Abstract

For a compact set $K\subset \mathbb {R}^1$ and a family $\{C_\lambda \}_{\lambda \in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\operatorname {dim}_{\rm H} K +\operatorname {dim}_{\rm H} C_\lambda \gt 1$ for all $\lambda \in J$, under natural technical conditions we prove that the sum $K+C_\lambda $ has positive Lebesgue measure for almost all values of the parameter $\lambda $. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than $1$.

Authors

  • Anton GorodetskiUniversity of California
    Irvine, CA 92697, U.S.A.
    e-mail
  • Scott NorthrupUniversity of California
    Irvine, CA 92697, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image