Dimensions of sums with self-similar sets
@article{Oberlin2015DimensionsOS, title={Dimensions of sums with self-similar sets}, author={Daniel Oberlin and Richard Oberlin}, journal={arXiv: Classical Analysis and ODEs}, year={2015} }
For some self-similar sets K in d-dimensional Euclidean space we obtain certain lower bounds for the lower Minkowski dimension of K+E in terms of the lower Minkowski dimension of E.
2 Citations
On arithmetic sums of fractal sets in Rd
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A compact set E⊂Rd is said to be arithmetically thick if there exists a positive integer n so that the n ‐fold arithmetic sum of E has non‐empty interior. We prove the arithmetic thickness of E , if…
On arithmetic sums of fractal sets in ${\Bbb R}^d$
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A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the…
References
SHOWING 1-9 OF 9 REFERENCES
Lower Bounds for Dimensions of Sums of Sets
- Mathematics
- 2008
We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝd.
The size of sums of sets, II
- Mathematics
- 1986
AbstractWe prove inequalities which give lower bounds for the Lebesgue measures of setsE +K whereK is a certain kind of Cantor set. For example, ifC is the Cantor middle-thirds subset of the circle…
Sums of Cantor sets
- MathematicsErgodic Theory and Dynamical Systems
- 1997
We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a…
Resonance between Cantor sets
- MathematicsErgodic Theory and Dynamical Systems
- 2009
Abstract Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two…
Additive combinatorics
- MathematicsCambridge studies in advanced mathematics
- 2007
The circle method is introduced, which is a nice application of Fourier analytic techniques to additive problems and its other applications: Vinogradov without GRH, partitions, Waring’s problem.
On the Dimension of Iterated Sumsets
- Mathematics
- 2009
Let A be a subset of the real line. We study the fractal dimensions of the k-fold iterated sumsets kA, defined as
$$kA =\{ {a}_{1} + \cdots + {a}_{k} : {a}_{i} \in A\}.$$
We show that for any…
E-mail address: roberlin@math.fsu.edu
- E-mail address: roberlin@math.fsu.edu
E-mail address: oberlin@math.fsu.edu
- E-mail address: oberlin@math.fsu.edu