The Speed of Convergence with respect to the Kolmogorov-Smirnov Metric in the Soshnikov Central Limit Theorem for the Sine-Process

For rescaled additive functionals of the sine-process, upper bounds are obtained for their speed of convergence to the Gaussian distribution with respect to the Kolmogorov-Smirnov metric. Under scaling with coefficient $R$ the Kolmogorov-Smirnov distance is bounded from above by $c\mathrm{/}\log R$ for a smooth function and by $c\mathrm{/}R$ for a function holomorphic in a horizontal strip.