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 (Ω, A, P), taking values in the lattice L(v₀,1)=ₖ=v₀+ k,k∈ Z\ and such that ϑX=∑k∈ Zf(k) f(k+1)>0, where f(k)=P=vₖ\, for each k. Let Xᵢ, i≥ 1 be independent, identically distributed random variables having the same law as X, and let Sₙ=∑ⱼ₌₁ⁿXⱼ, for each n. Let μ̄= \μₖ, k∈ Z\ be a sequence of non-negative reals such that 0<μₖ < f(k), if $f(k)>0$, μₖ=0 if $f(k)=0$, let μ= ∑k∈ Zμₖ, and let us assume that 1- μ< ϑX. Let also τ̄= \τₖ, k∈ Z\ be a sequence of non-negative reals satisfying the (solvable) equation: τₖ₋₁+τₖ2 =f(k) - μₖ, for all k ∈ Z. Let ϑ = ∑k∈ Z τₖ = 1- μ. Further let s(t) =∑k∈ Z μₖ e2i π vₖt and ρ be such that 1-ϑ<ρ<1. Let X be the random variable associated to X and μ̄, and defined by the relation P =vₖ\= τₖ/ϑ, for all k∈ Z. We prove the following uniform semi-local theorems for the class F=, d≥ 2\, where Fd= dN. {enumerate}[ (i)] There exists θ=θ(ρ,ϑ) with 0<θ <ϑ, C and N such that for n ≥ N,{multline*}u≥ 0d≥ 2 | P\{ S_n+u∈ Fd\} - 1/d-1/d∑0<||<d(e(iπ /d-π^2^2/2 d^2) ϑ Ee^{2i π /d{X}}+s (/d))^n |\\ ≤ {C}{θ3/2}{(log n)5/2}{n3/2}+2ρ^n.{multline*} Let D be a test set of divisors ≥ 2, D_φ be the section of D at height φ and |D_φ| denote its cardinality. Then,\[∑ₙ₌₂^∞ u≥ 0 φ≥ 2 {1øver |D_φ|} ∑d∈ D_φ | P \{d|S_n+u\} - 1/d| ≤ C_1/ϑ + {C_2}{θ3/2} + 2ρ^2/1-ρ,\]where C₁=2eπ²/4(1-e-π²/16), C₂= C∑ₙ₌₂^∞ (log n)5/2n3/2.{enumerate}