Unimodular solutions of infinite systems of linear equations

Donald Charles Benson

Vol. 15 (1965), No. 1, 1–11

DOI: 10.2140/pjm.1965.15.1

Abstract

It is well known that if a series of real numbers ${\sum }_{n=1}^{\infty }{a}_n$ converges, but not absolutely, then for any $b$, there exists a sequence $\left\{x_i\right\},x_i=\pm 1$, such that ${\sum }_{n=1}^{\infty }{a}_n{x}_n=b$. In §1, a criterion is given on a system of denumerably many equations of this type, with real coefficients, so that solutions $x_i=\pm 1$ exist for arbitrary right hand sides. A sequence $\left\{x_i\right\}$ such that $x_i=\pm 1$ will be called unimodular. In §2, there results are extended to finite systems, and it is shown that an infinite system has unimodular solutions for arbitrary right hand sides if and only if every finite subsystem has this property. §3 shows that if a system satisfies the criterion of §1, then, in a certain sense, “almost any” sequence $\left\{x_i\right\},x_i=\pm 1$, “satisfies” the system for any choice of right hand sides. In §4, conditions are given whereby infinite systems can be constructed which satisfy the criterion of §2. It follows, for example, that the system

$$\sum _{j=1}^{\infty }{\left(-1\right)}^{\left[j∕2^i\right]}{j}^{-\alpha }{x}_j=b_i,i=1,2,\cdots ;0<\alpha \leqq 1$$

has solutions $\left(x_i=\pm 1\right)$ for any $b_i\left(i=1,2,\cdots \right)$. The $b_i$ are allowed to be real numbers or $\pm \infty$.

Mathematical Subject Classification

Primary: 40.99

Milestones

Received: 22 January 1964

Published: 1 March 1965