  In this paper, we derive the improved uniform error bounds for the long-time
dynamics of the $d$-dimensional $(d=2,3)$ nonlinear space fractional
sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is
characterized by $\varepsilon^2$ where $0<\varepsilon \le 1$ is a dimensionless
parameter. The second-order time-splitting method is applied to the temporal
discretization and the Fourier pseudo-spectral method is used for the spatial
discretization. To obtain the explicit relation between the numerical errors
and the parameter $\varepsilon$, we introduce the regularity compensation
oscillation technique to the convergence analysis of fractional models. Then we
establish the improved uniform error bounds $O\left(\varepsilon^2
\tau^2\right)$ for the semi-discretization scheme and $O\left(h^m+\varepsilon^2
\tau^2\right)$ for the full-discretization scheme up to the long time at
$O(1/\varepsilon^2)$. Further, we extend the time-splitting Fourier
pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex
NSFSGE, and the improved uniform error bounds for them are also given. Finally,
extensive numerical examples in two-dimension or three-dimension are provided
to support the theoretical analysis. The differences in dynamic behaviors
between the fractional sine-Gordon equation and classical sine-Gordon equation
are also discussed.
