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13 July 2020
 
  » arxiv » math.NT/0404137

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A local-global theorem on periodic maps
Zhi-Wei Sun ;
Date 6 Apr 2004
Subject Number Theory; Combinatorics MSC-class: 05E99; 11A25; 11B25; 11B75; 20D60; 20F99 | math.NT math.CO
AbstractLet $psi_1,...,psi_k$ be maps from Z to an additive abelian group with positive periods $n_1,...,n_k$ respectively. We show that the function $psi=psi_1+...+psi_k$ is constant if $psi(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,...,n_s-1; s=1,...,k}; moreover, there are periodic maps $f_0,...,f_{|S|-1}$ from Z to Z only depending on S such that $psi(x)=sum_{r=0}^{|S|-1}f_r(x)psi(r)$ for all integers x. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187--196], and has various applications.
Source arXiv, math.NT/0404137
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