\langle   T_k f, g \rangle\rvert 
\leq \lVert T_k \;:\; \ell ^2 \to \ell ^2 \rVert \cdot  \langle f  \rangle _{I,2} \langle g \rangle _{I,2} \lvert  I\rvert ,$$ so it suffices to estimate the operator norm above. The assertion is that with high probability, the operator norm is small: $$\label{e:small}
\mathbb P  \bigl( \lVert T_k \;:\; \ell ^2 \to \ell ^2 \rVert  > C \sqrt k 2 ^{- k \frac {1- \alpha  } 2  }  \bigr) \lesssim  2 ^{-k},$$ provided $ C$ is sufficiently large. Combine this with the Borel Cantelli Lemma to prove the Lemma as stated.

By Plancherel’s Theorem, the operator norm is equal to $ \lVert Z (\theta )\rVert _{L ^{\infty } ( d\theta )}$, where $$Z (\theta ) :=  
\sum_{ n \;:\; 2 ^{k} \leq \lvert  n\rvert < 2 ^{k+1} }  Y_n \frac {   e ^{ 2 \pi i \theta }} {n ^{1- \alpha } }.$$ The expression above is a random Fourier series, with frequencies at most $ 2 ^{k+2}$. By Bernstein’s Theorem for trigonometric polynomials, the $ L ^{\infty } (d \theta )$ norm can be estimated by testing the norm on at most $ 2 ^{k+3}$ equally spaced points in $ \mathbb T $, 