Covering the integers by arithmetic sequences. II
Abstract: Let ( be a system of arithmetic sequences where and . For system will be called an (exact) -cover of if every integer is covered by at least (exactly) times. In this paper we reveal further connections between the common differences in an (exact) -cover of and Egyptian fractions. Here are some typical results for those -covers of : (a) For any there are at least positive integers in the form where . (b) When (, either or , and for each positive integer the binomial coefficient can be written as the sum of some denominators of the rationals if forms an exact -cover of . (c) If is not an -cover of , then have at least distinct fractional parts and for each there exist such that (mod 1). If forms an exact -cover of with or () then for every and there is an such that (mod 1).
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China; Dipartimento di Matematica, Università degli Studi di Trento, I-38050 Povo (Trento), Italy
Received by editor(s): June 7, 1994
Received by editor(s) in revised form: November 10, 1995
Additional Notes: This research is supported by the National Natural Science Foundation of P. R. China.
Article copyright: © Copyright 1996 American Mathematical Society