  In the context of incompressible fluids, the observation that turbulent
singular structures fail to be space filling is known as ``intermittency'' and
it has strong experimental foundations. Consequently, as first pointed out by
Landau, real turbulent flows do not satisfy the central assumptions of
homogeneity and self-similarity in the K41 theory, and the K41 prediction of
structure function exponents $\zeta_p=\frac{p}{3}$ might be inaccurate. In this
work we prove that, in the inviscid case, energy dissipation that is
lower-dimensional in an appropriate sense implies deviations from the K41
prediction in every $p-$th order structure function for $p>3$. By exploiting a
Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor's
frozen turbulence hypothesis, our strongest upper bound on $\zeta_p$ coincides
with the $\beta-$model proposed by Frisch, Sulem and Nelkin in the late 70s,
adding some rigorous analytical foundations to the model. More generally we
explore the relationship between dimensionality assumptions on the dissipation
support and restrictions on the $p-$th order absolute structure functions. This
approach differs from the current mathematical works on intermittency by its
focus on geometrical rather than purely analytical assumptions. The proof is
based on a new local variant of the celebrated Constantin-E-Titi argument that
features the use of a third order commutator estimate, the special double
regularity of the pressure, and mollification along the flow of a vector field.
