Falconer distance problem, additive energy and Cartesian products

Alex Iosevich; Bochen Liu

doi:10.5186/aasfm.2016.4135

Falconer distance problem, additive energy and Cartesian products

Alex Iosevich
, Bochen Liu

University of Rochester

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

A celebrated result due to Wolffsays if E is a compact subset of R², then the Lebesgue measure of the distance set Δ(E) = (|x - y|: x, y ∈ E) is positive if the Hausdorffdimension of E is greater than 4/3. In this paper we improve the 4/3 barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's d/2 + 1/3 exponent to d²/2d-1. The proof uses a combination of Fourier analysis and additive comibinatorics.

Original language	English
Pages (from-to)	579-585
Number of pages	7
Journal	Annales Academiae Scientiarum Fennicae Mathematica
Volume	41
DOIs	https://doi.org/10.5186/aasfm.2016.4135
State	Published - 2016
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2016, Annales Academiæ Scientiarum Fennicæ Mathematica.

Keywords

Additive energy
Ahlfors-David regular
Cartesian products
Distance problem

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10.5186/aasfm.2016.4135

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abstract = "A celebrated result due to Wolffsays if E is a compact subset of R2, then the Lebesgue measure of the distance set Δ(E) = (|x - y|: x, y ∈ E) is positive if the Hausdorffdimension of E is greater than 4/3. In this paper we improve the 4/3 barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's d/2 + 1/3 exponent to d2/2d-1. The proof uses a combination of Fourier analysis and additive comibinatorics.",

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AU - Liu, Bochen

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AB - A celebrated result due to Wolffsays if E is a compact subset of R2, then the Lebesgue measure of the distance set Δ(E) = (|x - y|: x, y ∈ E) is positive if the Hausdorffdimension of E is greater than 4/3. In this paper we improve the 4/3 barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's d/2 + 1/3 exponent to d2/2d-1. The proof uses a combination of Fourier analysis and additive comibinatorics.

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KW - Distance problem

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Falconer distance problem, additive energy and Cartesian products

Abstract

Bibliographical note

Keywords

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