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.

Abstract. We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

We consider a semiclassical (pseudo)differential operator on a compact surface (M, g), such that the Hamiltonian flow generated by its principal symbol admits a hyperbolic periodic orbit γ at some energy E 0 . For any ε > 0, we then explicitly construct families of quasimodes of this operator, satisfying an energy width of order ε | log | in the semiclassical limit, but which still exhibit a "strong scar" on the orbit γ, i.e. that these states have a positive weight in any microlocal neighbourhood of γ. We pay attention to optimizing the constants involved in the estimates. This result generalizes a recent result of Brooks [Br13] in the case of hyperbolic surfaces. Our construction, inspired by the works of Vergini et al. in the physics literature, relies on controlling the propagation of Gaussian wavepackets up to the Ehrenfest time.