The linear diophantine problem of Frobenius for subsets of arithmetic sequences

Abstract

Let A k = {a 1,. . . , a k } $ \subset \Bbb N $ with gcd (a 1,. . . , a k ) = 1. We shall say that a natural number n has a representation by a 1,. . . , a k if $ n =\sum \limits_{i=1}^{k}a_ix_i,\; x_i\in \Bbb N_0 $ . Let g = g (A k ) be the largest integer with no such representation. We then study the set A k = {a,ha + d,ha + 2d,..., ha + (k - 1) d} h,d > 0, gcd (a,d) = 1). If l k denotes the greatest number of elements which can be omitted without altering g (A k ), we show that ¶¶ $ 1-{4 \over \sqrt k} \le {l_k\over k} \le 1 - {3\over k}, $ ¶¶ provided a > k, or a = k with $ d \ge 2 h \sqrt {k} $ . The lower bound can be improved to 1 - 4 / k if we choose a > (k - 4) k + 3. Moreover, we determine sets $ E_k \subset A_k $ such that g (A k \ E k ) = g (A k ).