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Let Γ be a closed set in ${ℝ}^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_1>0$ and $c_2>0$ such that $c_1{r}^d\le \mu \left(B(x,r)\right)\le c_2{r}^d$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_p\left(\Gamma \right)$, 0 < p ≤ ∞, with respect to...