For any positive regularity parameter $\beta < 1/2$, we construct infinitely many weak solutions of the 3D incompressible Euler equations on the periodic box, which lie in $C^0_t H^\beta_x$. In particular, these solutions may be taken to have an $L^2$-based regularity index strictly larger than $1/3$, thus deviating from the scaling of the Kolmogorov-Obhukov $5/3$ power spectrum in the inertial range. This is a joint work with T. Buckmaster, N. Masmoudi, and M. Novack.