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Non-standard number representation: computer arithmetic, beta-numeration and quasicrystals

The purpose of this survey is to present the main concepts and results in non-standard number representation, and to give some examples of practical applications. This domain lies at the interface between discrete mathematics (dynamical systems, number theory, combinatorics) and computer science (computer arithmetic, cryptography, coding theory, algorithms). It also plays an important role in the modelization of physical structures like quasicrystals.