A remark on infinite arithmetic progressions

✍ Scribed by H. Lefmann; B. Voigt

• Publisher: Elsevier Science
• Year: 1984
• Tongue: English
• Weight: 208 KB
• Volume: 52
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(84)90089-x

✦ Synopsis

Let Se be a countably family of in&rite subsets of nonnegative integers. Using a diagonal argumpnt it is straight forward to select for each XE& an infinite subset S(X) r~ X such that the family (S(X) 1 XE Se) is disjoint. Also, there exists a mapping A :N --, N such that for every XE & and i EN there exists sorue i E X with A@ = i. However, in general the sets S(_X) cannot be obtained by a uniform procedure and consequently ihe mapy:ings S are not constructive.

In this short note we show that the situation is much more satisfying if s4 is the family of inkite arithmetic progressions.

An infinite arithmetic progression is a set A(u, 6) = (a + A

l b 1 A EN} of nonnegative integers, where a EN is nonnegatk~e and b E N(O) a positive integer. For subsets XEN of nonnegative integers let Ax(a, b) = (a + h l b \ A E x) be the X-subset of A(a, b).