Sharpness of Falconer’s (Figure Presented) Estimate

Abstract

In the paper introducing the celebrated Falconer distance problem, Falconer proved that the Lebesgue measure of the distance set is positive, provided that the Hausdorff dimension of the underlying set is greater than (Figure presented). His result is based on the estimate (0.1) (Figure presented), where µ is a Borel measure satisfying the energy estimate (Figure presented) for (Figure presented). An example due to Mattila [12, Remark 4.5], [11] shows in two dimensions that for no (Figure presented) does (Figure presented) imply (0.1). His construction can be extended to three dimensions. Mattila’s example readily applies to the case when the Euclidean norm in (0.1) is replaced by a norm generated by a convex body with a smooth boundary and non-vanishing Gaussian curvature. In this paper we prove, for all d ≥ 2, that for no (Figure presented) does Is(µ) < ∞ imply (0.1) or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body B with a smooth boundary and everywhere non-vanishing curvature. Our construction is based on a combinatorial construction due to Valtr [15].

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.5186/AASFM.2016.4145

Keywords

discrete geometric combinatorics, Geometric measure theory

Publication Date

1-1-2016

Journal Title

Annales Academiae Scientiarum Fennicae Mathematica

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