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We consider off-diagonal Jacobi matrices $J$ with (faster-than-exponential)sparse perturbations. We prove (Theorem \ref{onehalf}) that the Fouriertransform $\hat{\left\| f\right\| ^{2}d\rho}(t)$ of the spectral measure $\rho$ of $J$, whose sparse perturbations are at least separated by a distance $\exp\left(cj(\ln j)^{2}\right) /\delta ^{j}$, for some $c>1/2,$ $0<\delta <1$ andfor a dense subset of $C_{0}^{\infty}(-2,2)$-functions $f$, decays as$t^{-1/2}\Omega (t)$, uniformly in the spectrum $[-2,2]$, $\Omega (t)$increasing less rapidly than any positive power of $t$, improving earlierresults obtained by Simon (Commun. Math. Phys. \textbf{179}, 713-722 (1996))and by Krutikov-Remling (Commun. Math. Phys. \textbf{223}, 509-532 (2001)) forSchr\"{o}dinger operators with sparse potential that increases as fast asexponential-of-exponential. Applications to the spectrum of the Kronecker sumof two (or more) copies of the model are given.

Pointwise Decay of Fourier-Stieltjes transform of the Spectral Measure for Jacobi Matrices with Faster-than-Exponential Sparse Perturbations