On the Structure of Periodic Arithmetical Maps
نویسندگان
چکیده
This is an introduction to the algebraic theory of periodic arithmetical maps. The topic is connected with number theory, combinatorics, algebra and analysis. 1. Various Backgrounds I. Combinatorial Background. For n ∈ Z = {1, 2, 3, · · · } and a ∈ R(n) = {0, 1, · · · , n− 1}, let a(n) = a+ nZ = {x ∈ Z : x ≡ a (mod n)} = {· · · , a− n, a, a+ n, · · · } and call it a residue class with modulus n. For a finite system (1.1) A = {as(ns)}s=1 of such residue classes, we define its covering function wA : Z → N = {0, 1, 2, · · · } by (1.2) wA(x) = |{1 ≤ s ≤ k:x ≡ as (mod ns)}|. If wA(x) ≥ m for all x ∈ Z, then we call A an m-cover of Z; if wA(x) = m for all x ∈ Z, then we call A an exact m-cover of Z. An exact 1-cover is also called a Typeset by AMS-TEX 1
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Algebraic Approaches to Periodic Arithmetical Maps
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