  Onsager's conjecture states that the conservation of energy may fail for $3D$
incompressible Euler flows with H\"{o}lder regularity below $1/3$. This
conjecture was recently solved by the author, yet the endpoint case remains an
interesting open question with further connections to turbulence theory. In
this work, we construct energy non-conserving solutions to the $3D$
incompressible Euler equations with space-time H\"{o}lder regularity converging
to the critical exponent at small spatial scales and containing the entire
range of exponents $[0,1/3)$.
  Our construction improves the author's previous result towards the endpoint
case. To obtain this improvement, we introduce a new method for optimizing the
regularity that can be achieved by a convex integration scheme. A crucial point
is to avoid power-losses in frequency in the estimates of the iteration. This
goal is achieved using localization techniques of \cite{IOnonpd} to modify the
convex integration scheme.
  We also prove results on general solutions at the critical regularity that
may not conserve energy. These include a theorem on intermittency stating
roughly that energy dissipating solutions cannot have absolute structure
functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their
singular supports have space-time Lebesgue measure zero.
