A uniform semi-local limit theorem along sets of multiples for sums of i.i.d. random variables

Let $X$ be a square integrable random variable with basic probability space $(\text{Ø},\text{A},\mathrm{\P })$, taking values in a lattice $\mathcal{L}(v_0,1)=\{v_k=v_0+k,k\in \mathbb{Z}\}$ and such that ${\text{t}}_X={\sum }_{k\in \mathbb{Z}}\mathrm{\P }\{X=v_k\}\wedge \mathrm{\P }\{X=v_{k+1}\}>0$. Let $X_i$, $i\ge 1$ be independent, identically distributed random variables having same law than $X$, and let $S_n={\sum }_{j=1}^n{X}_j$, for each $n$. Let ${\text{m}}_k\ge 0$ be such that $\text{m}={\sum }_{k\in \mathbb{Z}}{\text{m}}_k$ verifies $1-{\text{t}}_X<\text{m}<1$, noting that ${\text{t}}_X<1$ always. Further let $\text{t}=1-\text{m}$, $s(t)={\sum }_{k\in \mathbb{Z}}{\text{m}}_k{e}^{2i\pi v_{k}t}$ and $\rho$ be such that $1-\text{t}<\rho <1$. We prove the following uniform semi-local theorems for the class $\mathcal{F}=\{F_d,d\ge 2\}$, where $F_d=d\mathbb{N}$. \noi(i) There exists $\theta =\theta (\rho ,\text{t})$ with $0<\theta <\text{t}$, $C$ and $N$ such that for $n\ge N$, \begin{align*} \sup_{u\ge 0},\sup_{d\ge 2} \Big| ¶{ S_n+u\in F_{d} } - {1\over d}- {1\over d}\sum_{ 0< |\ell|<d }& \Big( e^{ (iπ{\ell\over d }-{ π^2\ell^2\over 2 d^2}) } \t, \E ,e^{2i π{\ell\over d }\widetilde X } +s\big( {\ell\over d }\big)\Big)^n \Big| \cr &\le \frac{C }{ θ^{3/2}}\ \frac{(\log n)^{5/2}}{ n^{3/2}}+2ρ^n. \end{align*} \vskip 1 pt \noi(ii) Let $\mathcal{D}$ be a test set of divisors $\ge 2$, ${\mathcal{D}}_{\text{p}}$ be the section of $\mathcal{D}$ at height $\text{p}$ and $\mathrm{\mid }{\mathcal{D}}_{\text{p}}\mathrm{\mid }$ denote its cardinality. Then, \begin{eqnarray*} \sum_{n=N}^\infty \ \sup_{u\ge 0} , \sup_{\p\ge 2}, {1\over |\mathcal D_\p |} \sum_{d\in \mathcal D_\p } ,\Big| ¶{d|S_n+u } - {1\over d}\Big| & \le & \frac{C_1}{\t} , + \frac{C_2 }{ θ^{3/2}} +\frac{2ρ^2}{1-ρ}. \end{eqnarray*}