On angles determined by fractal subsets of the Euclidean space

Abstract

We prove that if the Hausdorff dimension of a compact subset of R d is greater than d+1 2 , then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for bi-linear analogs of generalized Radon transforms and the method of stationary phase play a key role. These results complement those of V. Harangi, T. Keleti, G. Kiss, P. Maga, P. Mattila and B. Stenner in ([8]). We also obtain new upper bounds for the number of times an angle can occur among N points in R d , d ≥ 4, motivated by the results of Apfelbaum and Sharir ([1]) and Pach and Sharir ([13]). We then use this result to establish sharpness results in the continuous setting. Another sharpness result relies on the distribution of lattice points on large spheres in higher dimensions.

Finite chains inside thin subsets of ℝd

“…We already established using Theorem [9] that finite compositions of the operators T l applied to L 2 (µ) functions are in L 2 (µ). Using the Cauchy-Schwarz inequality and in light of Lemma 2.5, we M k (t 1 + h 1 , t 2 + h 2 , · · · , t k + h k ) is bounded.…”

“…× E. Theorem 1.8 gives an upper bound on this expression that is independent of . This is accomplished using an inductive argument on the chain length coupled with repeated application of an earlier result from [9] in which the authors establish L 2 (µ) mapping properties of certain convolution operators. This upper bound is important in the final section where we define a measure on the set of chains.…”

“…We shall repeatedly use the following result due to Iosevich, Sawyer, Taylor, and Uriarte-Tuero [9].…”

Zeta functions and solutions of Falconer-type problems for self-similar subsets of ℤn

“…In particular, it would seemingly be more natural to work with simplices and replace distance by volume or configuration sets. The articles [9][10][11]17] have all been motivated to do just this by formulating and proving variants of Falconer-type problems, for example, for Vol(E) = {vol(Σ n (x)) : x = (x 1 , . .…”

“…As was noted in [17], little is known, in general, about the distribution of angles determined by points in unbounded discrete subsets of R n once n 4. Whereas the results proved in [17] require the hypothesis of s-adaptability, we give a solution to a Falconer problem for angles for any compatible self-similar subset of Z n once n 4 with no other hypothesis. The statement is given in Theorem 2 and proved in § 3.2.…”