  The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is
realized through a sequence of substitutions on an alphabet of three symbols;
particular attention is paid to $x=0$. The first application is to show that
any asymptotic growth rate of the discrepancy sums not trivially forbidden may
be achieved. A second application is to show that for badly approximable
$\theta$ and any $x$ the range of values taken over $i=0,1,...n-1$ is
asymptotically similar to $\log(n)$, a stronger conclusion than given by the
Denjoy-Koksma inequality.
