arXiv:2605.13347v2 Announce Type: replace Abstract: We establish sharp estimates for the discrete optimal constant of the fractional Sobolev inequality in dimension $N\geq 1$, with fractional exponent $s\in (0,\min\{1,N/2\})$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the problem is posed in the unit ball, for which we employ a quasi-uniform and regular mesh. Since the infimum defining the fractional Sobolev constant is not attained on bounded domains, these estimates quantify how far a piecewise affine mesh of size $h$ can resolve the concentration of the optimizers: the discrete minimizers are close to bubbles concentrated at the scale $h^{2(2-s)/(N+4(1-s))}$, and the energy gap is exactly of order $h^{2(2-s)(N-2s)/(N+4(1-s))}$.