Shifting the ordinates of zeros of the Riemann zeta function

Let $y\mathrm{\ne }0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_{\ast }>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_{\ast }$, both $$\zeta (\frac{1}{2}+i\gamma )=0\text{and}\zeta (\frac{1}{2}+i(\gamma +y))\mathrm{\ne }0$$ hold for at least one $\gamma$ in the interval $[T,T(1+\epsilon )]$, where $\epsilon :=T^{-C\mathrm{/}\log \log T}$.