Multilinear generalized Radon transforms and point configurations

Abstract

Abstract We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving (k+1)-point configurations in geometric measure theory, with k ≥ 2, including the distribution of simplices, volumes and angles determined by the points of fractal subsets E ⊂ ℝ d , d ≥ 2. If Tk (E) denotes the set of noncongruent ( k + 1 ) ${(k+1)}$ -point configurations determined by E, we show that if the Hausdorff dimension of E is greater than d - (d-1)/(2k), then the k + 1 2 ${\binom{k+1}{2}}$ -dimensional Lebesgue measure of Tk (E) is positive. This complements previous work on the Falconer conjecture ([Int. Math. Res. Not. IMRN 23 (2005), 1411–1425] and the references there), as well as work on finite point configurations [Recent Advances in Harmonic Analysis and Applications, Springer-Verlag, New York (2013), 93–103; Anal. PDE 5 (2012), no. 2, 397–409]. We also give applications to Erdős-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in [Adv. Math. 228 (2011), 2385–2402].