On Rozanov's theorem and strenghtened asymptotic uniform distribution

For sums Sₙ=∑ₖ₌₁ⁿ Xₖ, n≥ 1 of independent random variables Xₖ taking values in we prove, as a consequence of a more general result, that if (i) For some function 1≤ φ(t)↑ ∞ as t→ ∞, and some constant C, we have for all n and ν∈, {equation*}{abstract1} |B_n\{ S_n=ν\}- {1 √ 2π }\ e^{- {(ν-M_n)^2 2 B_n^2} }|\,≤ \, {C \,φ(B_n)}, {equation*} then (ii) There exists a numerical constant C₁, such that for all n such that Bₙ≥ 6, all h≥ 2, and =0,1,…, h-1, {align*}{abstract1} |{ P}\{ S_n≡\, \ { (mod h)}\}- 1/h| ≤ {1 √2π\, B_n }+{1+ 2 {C}/{h} }{ φ(B_n)2/3 } + C_1 \,e^{-(1/ 16 )φ(B_n)2/3}. {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 ⱼ =∑k∈ Pⱼ= k\ Pⱼ= k+1 \ >0, for each j and that νₙ =∑ⱼ₌₁ⁿ ⱼ↑ ∞. We prove also strenghtened forms of the asymptotic uniform distribution property.