## Project Details

### Description

The project "Fractals, digital expansions, words and random structures" lies at the intersection of several fields of

mathematics. The first part is devoted to fractals: the Sierpinski carpet and generalisations, and limiting net sets.

The latter were introduced by the applicant, have the property of being well distributed'' in a certain sense and

can be viewed as random fractals but also as generalised Sierpinski carpets. For limiting net sets we are interested,

e.g., in aspects like topological and fractal geometric properties, geodesic distances between points, percolation,

properties of their pore structure in the context of modelling porous materials with generalised Sierpinski carpets.

We are also interested in generalisations of limiting net sets.

The second part is dedicated to digital expansions, random words, and connections between them. Probability

measures that are inductively defined on the unit interval and stem from the study of arithmetic functions, like the

multinomial or the Gray code measure, induce probability measures on certain sets of random words, including

some distributions corresponding to digital expansions with missing digits. One of our aims is to extend results

already obtained (by the applicant or other authors) on these measures, and consider generalisations thereof. We

intend to apply these types of constructions also in higher dimensions, and to construct and study measures whose

support are fractals, e.g. some of those studied in the first part. Further problems regard combinatorics on words,

Hamming weights, and discrepancies of point sets in the unit interval, also with respect to special measures.

mathematics. The first part is devoted to fractals: the Sierpinski carpet and generalisations, and limiting net sets.

The latter were introduced by the applicant, have the property of being well distributed'' in a certain sense and

can be viewed as random fractals but also as generalised Sierpinski carpets. For limiting net sets we are interested,

e.g., in aspects like topological and fractal geometric properties, geodesic distances between points, percolation,

properties of their pore structure in the context of modelling porous materials with generalised Sierpinski carpets.

We are also interested in generalisations of limiting net sets.

The second part is dedicated to digital expansions, random words, and connections between them. Probability

measures that are inductively defined on the unit interval and stem from the study of arithmetic functions, like the

multinomial or the Gray code measure, induce probability measures on certain sets of random words, including

some distributions corresponding to digital expansions with missing digits. One of our aims is to extend results

already obtained (by the applicant or other authors) on these measures, and consider generalisations thereof. We

intend to apply these types of constructions also in higher dimensions, and to construct and study measures whose

support are fractals, e.g. some of those studied in the first part. Further problems regard combinatorics on words,

Hamming weights, and discrepancies of point sets in the unit interval, also with respect to special measures.

Status | Finished |
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Effective start/end date | 1/04/08 → 31/08/13 |