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Project ID: 306494
Funded under: FP7-IDEAS-ERC
Country: Israel

Mid-Term Report Summary - FRACTALSANDMETRICNT (Fractals, algebraic dynamics and metric number theory)

The project is motivated by number-theoretic conjectures concerning the complexity of the expansions of numbers in different bases, but focuses primarily on some related problems about the behavior of "fractal" sets and measures under certain geometric operations, e,g, taking the "sum" of the sets or scaled copies of the sets, or the intersection of scaled and translated copies of them. It is expected that fractal sets and measures arising in algebraic or number-theoretic contexts should behave more regularly under these operations than general fractal sets, in the sense that the typical behavior should be observed in all specific instances except when an algebraic obstruction is present. In particular there are precise predictions on how large (in terms of dimension) the results of these operations should be.

In the first period the project has achieved major progress on some of these problems, specifically on linear projections ("shadows") of certain fractal sets and the so-called "Bernoulli convolution" problem, resolving some cases of the first and advancing our understanding of the second. This was made possible by the successful application of ideas from "additive combinatorics" to fractal geometry. These results have found numerous applications, both in pure mathematics and in information theory.

The project has also produced several theoretical studies which deepen our understanding of the process of "zooming in" to fractals (via the so-called scenery flow), and a variety of results have been obtained by team members with applications to geometric measure theory (the general study of fractal measures) and Diophantine approximation (the study of approximation properties of general numbers by fractions).

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