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Article Dans Une Revue Mathematics Newsletter Année : 2022

Notes on the domain of exponent pairs

Résumé

The theory of exponent pairs as initiated by Phillipps in 1933 proposes pairs of exponents pκ, λq so that one has ř n"N e 2iπφpnq !ε F κ`ε N λ`ε , for any positive ε, where φ is a 'monomial-like' smooth function whose first derivative is of size about F. We propose to explore the domain of available pairs pκ, λq through a very geometrical approach. We prove in particular that this domain is the convex hull of a connected curve in the classical case. We also show that a possible choice for λ, for any κ P r0, 1{2s, is given by λ " 1 ´κ log 2 log 2κ`1 2κ. We finally recall rapidly how this theory has been adapted to the higher dimensional setting. In passing, we take the opportunity of this slow-paced paper to describe some usage of the SageMath software.
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hal-04467274 , version 1 (20-02-2024)

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Paternité - Pas d'utilisation commerciale

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  • HAL Id : hal-04467274 , version 1

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Olivier Ramaré, Julien Cassaigne, Sary Aurélien Drappeau. Notes on the domain of exponent pairs. Mathematics Newsletter, 2022, 33 (3 -- 4), pp.7--15. ⟨hal-04467274⟩
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