Application of a Fourier Restriction Theorem to Certain Families of Projections in $${\mathbb {R}}^3$$ R 3

Oberlin, Daniel M.; Oberlin, Richard

doi:10.1007/s12220-014-9480-7

Cited by 28 publications

(42 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first results for Hausdorff dimension in the general situation were, soon afterwards, obtained by D. and R. Oberlin [11]. Here is their main result: [11].) Let B ⊂ R 3 be an analytic set with dim B ≥ 1.…”

Section: Introductionmentioning

confidence: 97%

Hausdorff dimension estimates for restricted families of projections in R3

Orponen

2015

Advances in Mathematics

View full text Add to dashboard Cite

This paper is concerned with restricted families of projections in R 3 . Let K ⊂ R 3 be a Borel set with Hausdorff dimension dim K = s > 1. If G is a smooth and sufficiently wellcurved one-dimensional family of two-dimensional subspaces, the main result states that there exists σ(s) > 1 such that dim π V (K) ≥ σ(s) for almost all V ∈ G. A similar result is obtained for some specific families of one-dimensional subspaces.

show abstract

Section: Introductionmentioning

confidence: 97%

Hausdorff dimension estimates for restricted families of projections in R3

Orponen

2015

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…We fall short of proving the conjecture in two ways: first, we are only able to obtain a non-trivial lower bound for the packing dimension dim p (see [12, §5.9]) of the projections, and, second, our bound is much weaker than the full dimension conservation conjectured above (the first shortcoming has already been partially overcome in later work, see [13] and [16]). Our first main result is the following: Theorem 1.7.…”

mentioning

confidence: 99%

On restricted families of projections in ℝ³

Fässler

Orponen

2014

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

ABSTRACT. We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 1/2-dimensional sets B ⊂ R 3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε, proving that if dim H B = s > 1/2, then the packing dimension of the projections is almost surely at least σ(s) > 1/2. For projections onto planes, we obtain a similar bound, with the threshold 1/2 replaced by 1. In the special case of self-similar sets K ⊂ R 3 without rotations, we obtain a full Marstrand type projection theorem for oneparameter families of projections onto lines. The dim H K ≤ 1 case of the result follows from recent work of M. Hochman, but the dim H K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely. CONTENTS

show abstract

“…The conjectured lower bound is min{dim H E, 2} and the bound min{dim H E, 1} for all values of dim H E was obtained in [26]. The further improvements stated below come from Fourier restriction methods [65].…”

Section: Projections In Restricted Directionsmentioning

confidence: 91%

“…[26,65] Let E ⊂ R 3 be a Borel or analytic set, let θ(t) be a non-degenerate family of directions, and let proj V θ (t) denote projection onto the plane perpendicular to direction θ. Then, for almost all t ∈ (0, 1),…”

Section: Projections In Restricted Directionsmentioning

confidence: 99%

Fractal Geometry and Stochastics V

2015

Progress in Probability

View full text Add to dashboard Cite

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.

show abstract

Application of a Fourier Restriction Theorem to Certain Families of Projections in $${\mathbb {R}}^3$$ R 3

Abstract: We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R 3 whose normal directions form nondegenerate curves.

Cited by 28 publications

References 9 publications

Hausdorff dimension estimates for restricted families of projections in R3

Hausdorff dimension estimates for restricted families of projections in R3

On restricted families of projections in ℝ³

Fractal Geometry and Stochastics V

Contact Info

Product

Resources

About

Application of a Fourier Restriction Theorem to Certain Families of Projections in $${\mathbb {R}}^3$$ R 3

Abstract: We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R 3 whose normal directions form nondegenerate curves.

Cited by 28 publications

References 9 publications

Hausdorff dimension estimates for restricted families of projections in R3

Hausdorff dimension estimates for restricted families of projections in R3

On restricted families of projections in ℝ3

Fractal Geometry and Stochastics V

Contact Info

Product

Resources

About

On restricted families of projections in ℝ³