Fourier integral operators, fractal sets, and the regular value theorem

Abstract

We prove that if E ⊂ R 2d , for d ≥ 2, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by dim H (E), and φ is a sufficiently regular function, then the upper Minkowski dimension of the setin line with the regular value theorem from the elementary differential geometry. Our arguments are based on the mapping properties of the underlying Fourier integral operators and are intimately connected with the Falconer distance conjecture in geometric measure theory. We shall see that our results are, in general, sharp in the sense that if the Hausdorff dimension is smaller than a certain threshold, then the dimensional inequality fails in a quantifiable way. The constructions used to demonstrate this are based on the distribution of lattice points on convex surfaces and have connections with combinatorial geometry.

Improvement on 2-Chains Inside Thin Subsets of Euclidean Spaces

“…(see e.g. [3,7,6,8,9,12] and references therein). An interesting fact is, among all currently known results, the dimensional exponent d+1 2 appears again and again.…”

Intersections of sets and Fourier analysis

“…almost everywhere with respect to the probability measure dt dµ(x) and t ∈ [1,2]. Finally, for almost every t ∈ [1, 2],…”

“…for any N ≥ 1. Because we are assuming that |ξ| > 1 ǫ and because ψ(t) = 0 outside of [0.5, 2.5], we see that ǫt|ξ| ≥ 1 2 ǫ|ξ| on [1,2], and so we have the upper bound on (5.13) over the indicated region: 6. Appendix 6.1.…”

“…Interesting arithmetic issues arise in the consideration of sharpness examples. These results are partly motivated by those in [1] and [6] where in the former the classical regular value theorem from differential geometry was investigated in a fractal setting, and in the latter discrete incidence theory is explored from an analytic standpoint.Institute des HautesÉtudes Scientifiques, 35 route des Chartres,…”

Pinned distance problem, slicing measures, and local smoothing estimates

“…In [3], Eswarathasan, Iosevich and Taylor prove that if Φ : R d × R d → R has non-zero Monge-Ampere determinant (also called Phong-Stein rotational curvature condition), i.e.…”