A generalization of Marstrand's theorem for projections of cartesian products

  • Jorge Erick López

    IMPA, Estrada D. Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil
  • Carlos Gustavo Moreira

    IMPA, Estrada D. Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

Abstract

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets:

Let be Borel subsets of respectively, and be a surjective linear map. We set

Consider the space with the natural measure and set . For every and every we define . Then we have

Theorem.

(i) If , then has positive k-dimensional Lebesgue measure for almost every .

(ii) If and , then for almost every .

Cite this article

Jorge Erick López, Carlos Gustavo Moreira, A generalization of Marstrand's theorem for projections of cartesian products. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 4, pp. 833–840

DOI 10.1016/J.ANIHPC.2014.04.002