Reconstruction of rational polytopes from the real-parameter Ehrhart function of its translates

When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon; in particular, it is shown that, for full-dimensional $P$, not only there are infinitely many different functions $L_{P+w}(t)$ (for integer $w$), but that for rational $P$ the collection of these functions identifies $P$ uniquely.