Let \(Z\) be the set of all positive numbers \(\alpha\) with the following property: “For every non-negative integer \(n\), \[ 0\le (3/2)^n\alpha -[(3/2)^n\alpha] < 1/2." \] This note makes an attempt to decide whether \(Z\) has, or has not, any elements, but arrives only at partial results. It is, in particular, proved that \(Z\) is at most enumerable; that every interval between consecutive integers contains at most one element of \(Z\); and that for sufficiently large \(x\) there are fewer than \(x^{3/4}\) elements of \(Z\) between \(0\) and \(x\).