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
$$\sum _{j=1}^{\infty}{\left(1\right)}^{\left[j\u2215{2}^{i}\right]}{j}^{\alpha}{x}_{j}={b}_{i},\phantom{\rule{1em}{0ex}}i=1,2,\cdots \phantom{\rule{0.3em}{0ex}};\phantom{\rule{0.3em}{0ex}}0<\alpha \leqq 1$$
has solutions
$\left({x}_{i}=\pm 1\right)$
for any
${b}_{i}\left(i=1,2,\cdots \phantom{\rule{0.3em}{0ex}}\right)$. The
${b}_{i}$ are allowed to be
real numbers or
$\pm \infty $.
