Final Report Summary - FRACTALSANDMETRICNT (Fractals, algebraic dynamics and metric number theory)
Many fundamental mathematical objects are "fractal" in nature, meaning that small pieces look the same as the whole, up to a change in scale and possibly coordinates (stretching/rotation, etc.). Because of the repetition at infinitely small scales, it is often difficult to establish how large such objects are, e.g in terms of their Hausdorff dimension. This project has focused on some fundamental problems in this area, and specifically on some conjectures that predict that such objects should be maximally large, or maximally small, compared to the baseline prediction (which often is known to hold in some average sense). The project has accomplished many of its objectives; specifically, our work has lead to the resolution of some deep problems on the dimension of self-similar sets and their projections ("shadows" of the sets), as well as forming a major step in the recent resolution of the main conjecture on intersections (Furstenberg's intersection problem). The project has also resolved longstanding problems on the dimension of self-affine sets, i.e. sets in which not only scale but also the coordinate system changes at small scales.