On Rozanov's Theorem and strenghtened asymptotic uniform distribution

For sums $S_n={\sum }_{k=1}^n{X}_k$, $n\ge 1$ of independent random variables $X_k$ taking values in $\mathbb{Z}$ we prove, as a consequence of a more general result, that if (i) For some function $1\le \phi (t)\uparrow \mathrm{\infty }$ as $t\to \mathrm{\infty }$, and some constant $C$, we have for all $n$ and $\nu \in \mathbb{Z}$, \begin{equation*}\label{abstract1} \big|B_n¶\big{ S_n=ν\big}- {1\over \sqrt{ 2π} }\ e^{- {(ν-M_n)^2\over 2 B_n^2} }\big|,\le , {C\over ,φ(B_n)}, \end{equation*} then (ii) There exists a numerical constant $C_1$, such that for all $n$ such that $B_n\ge 6$, all $h\ge 2$, and $\text{m}=0,1,\dots ,h-1$, \begin{align*}\label{abstract1} \Big|{\mathbb P}\big{ S_n\equiv, \m\ \hbox{\rm{ (mod $h$)}}\big}- \frac{1}{h}\Big| \le {1\over \sqrt{2π}, B_n }+\frac{1+ 2 {C}/{h} }{ φ(B_n)^{2/3} } + C_1 ,e^{-(1/ 16 )φ(B_n)^{2/3}}. \end{align*} Assumption (i) holds if a local limit theorem in the usual form is applicable, and (ii) yields a strenghtening of Rozanov’s necessary condition. Assume in place of (i) that ${\text{t}}_j={\sum }_{k\in \mathbb{Z}}\mathbb{P}\{X_j=k\}\wedge \mathbb{P}\{X_j=k+1\}>0$, for each $j$ and that ${\nu }_n={\sum }_{j=1}^n{\text{t}}_j\uparrow \mathrm{\infty }$. We prove also strenghtened forms of the asymptotic uniform distribution property.