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Veech's Theorem claims that if $G$ is a locally compact\,(LC) Hausdorfftopological group, then it may act freely on $G^{\textrm{LUC}}$. We proveVeech's Theorem for $G$ being only locally quasi-totally bounded, notnecessarily LC. And we show that the universal a.a. flow is the maximal almost1-1 extension of the universal minimal a.p. flow and is unique up to almost 1-1extensions. In particular, every endomorphism of Veech's hull flow induced byan a.a. function is almost 1-1; for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$ actsfreely on its canonical universal a.a. space. Finally, we characterize Bochnera.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$ (due toVeech 1965 for the special case that $G$ is abelian, LC, $\sigma$-compact, andfirst countable).

Veech's Theorem of $G$ acting freely on $G^{\textrm{LUC}}$ and Structure Theorem of a.a. flows