BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-72-275
       ENTRY:: October 16, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Sets generated by iteration of a linear operation.
        TYPE:: Technical Report
      AUTHOR:: Klarner, David A.
        DATE:: March 1972
       PAGES:: 11
    ABSTRACT:: This note is a continuation of the paper "Arithmetic
               properties of certain recursively defined sets," written in
               collaboration with Richard Rado. Here the sets under
               consideration are those having the form S = <$m_1 x_1 +
               \ldots\ + m_r x_r : 1$> where $m_1,\ldots ,m_r$ are given
               natural numbers with greatest common divisor 1. The set S is
               the smallest set of natural numbers which contains 1 and is
               closed under the operation $m_1 x_1 + \ldots\ + m_r x_r$.
               Also, S can be constructed by iterating the operation $m_1
               x_1 + \ldots\ + m_r X_r$ over the set {1}. For example, <2x +
               3y: 1> = {1, 5, 13, 17, 25, $\ldots\ } = (1 + 12N) $\cup$ (5
               + 12N) where N = {0,1,2,$\ldots$}. It is shown in this note
               that S contains an infinite arithmetic progression for all
               natural numbers r-1,$m_1,\ldots ,m_r$. Furthermore, if
               ($m_1,\ldots ,m_r$) = ($m_1\ldots m_r,m_1 + \ldots\ + m_r$) =
               1, then S is a per-set; that is, S is a finite union of
               infinite arithmetic progressions. In particular, this implies
               <mx + ny: 1> is a per-set for all pairs {m,n} of relatively
               prime natural numbers. It is an open question whether S is a
               per-set when ($m_1,\ldots ,m_r$) = 1, but ($m_1\ldots m_r,m_1
               + \ldots\ + m_r$) > 1.
       NOTES:: [Adminitrivia V1/Prg/19951016]
         END:: STAN//CS-TR-72-275