### Description

Systems of numeration give different ways of representing the positive integers as weighted sums; special and

classical examples are the decimal expansion (203=2x100+0x10+3x1) or the binary expansion used by arithmetical

processors. Alternative expansions allowing either signed digits or more complicated bases than 1,10,100,... have

also been studied and used in cryptography, analysis of algorithms, and game theory.

The general theory of such systems has its origin in theoretical computer science, where they were studied

extensively mostly for speeding up computations. We refer to Knuth's books on the Art of Computer Programming.

In the proposed project we take a different point of view, which was initiated by Vershik and Liardet: the

representations of the integers are embedded in a topological space. The addition of 1 is used to define a dynamical

system called the odometer. Through the somehow intricate structure of the carry propagation this system reflects

the combinatorial properties of the underlying number representation. We intend to develop this theory in a general

setting and to extend it to other sets such as number fields or polynomial rings.

The probabilistic study of classical arithmetical functions has been developed in order to achieve a better

understanding of the statistical structure of the prime decomposition of integers. Probabilistic number theory for

arithmetical functions related to systems of numeration shall be developed during the course of this project. The

space constructed above will play the rle of several compactifications of the integers that have been used for

classical arithmetical functions.

classical examples are the decimal expansion (203=2x100+0x10+3x1) or the binary expansion used by arithmetical

processors. Alternative expansions allowing either signed digits or more complicated bases than 1,10,100,... have

also been studied and used in cryptography, analysis of algorithms, and game theory.

The general theory of such systems has its origin in theoretical computer science, where they were studied

extensively mostly for speeding up computations. We refer to Knuth's books on the Art of Computer Programming.

In the proposed project we take a different point of view, which was initiated by Vershik and Liardet: the

representations of the integers are embedded in a topological space. The addition of 1 is used to define a dynamical

system called the odometer. Through the somehow intricate structure of the carry propagation this system reflects

the combinatorial properties of the underlying number representation. We intend to develop this theory in a general

setting and to extend it to other sets such as number fields or polynomial rings.

The probabilistic study of classical arithmetical functions has been developed in order to achieve a better

understanding of the statistical structure of the prime decomposition of integers. Probabilistic number theory for

arithmetical functions related to systems of numeration shall be developed during the course of this project. The

space constructed above will play the rle of several compactifications of the integers that have been used for

classical arithmetical functions.

Status | Finished |
---|---|

Effective start/end date | 1/10/05 → 31/12/05 |