A connection between covers of the integers and unit fractions

作者：Sun Zhi Wei^*

来源：Advances in Applied Mathematics, 2007, 38(2): 267-274.

DOI：10.1016/j.aam.2006.02.002

For integers a and n > 0, let a(n) denote the residue class {x is an element of Z: x a (mod n)}. Let A be a collection {a(s)(n(s))}(s=1)(k) of finitely many residue classes such that A covers all the integers at least m times but {a(s)(n(s))}(s=1)(k-1) does not. We show that if n(k) is a period of the covering function w(A)(x) = vertical bar{1 <= s <= k: x is an element of a(s)(n(s))}vertical bar then for any r = 0,...,n(k) - 1 there are at least m integers in the form Sigma(s is an element of l) 1/n(s) - r/n(k) with I subset of {1,..., k-1}.

• 出版日期2007-2
• 单位南京大学