  We investigate the viability of sub-millimeter wavelength oscillating
deviations from the Newtonian potential at both the theoretical and the
experimental/observational level. At the theoretical level such deviations are
generic predictions in a wide range of extensions of General Relativity (GR)
including $f(R)$ theories, massive Brans-Dicke theories, compactified extra
dimension models and nonlocal extensions of GR. However, the range of
parameters associated with such oscillating deviations is usually connected
with instabilities. An exception emerges in nonlocal gravity theories where
oscillating deviations from Newtonian potential occur naturally on
sub-millimeter scales without instabilities. As an example of a model with
unstable Newtonian oscillations we review an $f(R)$ expansion around General
Relativity of the form $f(R)=R+\frac{1}{6 m^2} R^2$ with $m^2<0$ pointing out
possible stabilization mechanisms. As an example of a model with stable
Newtonian oscillations we discuss nonlocal gravity theories. If such
oscillations are realized in Nature on sub-millimeter scales, a signature is
expected in torsion balance experiments testing Newton's law. We search for
such a signature in the torsion balance data of the Washington experiment
(combined torque residuals) testing Newton's law at sub-millimeter scales. We
show that an oscillating residual ansatz with spatial wavelength $\lambda
\simeq 0.1mm$ provides a better fit to the data compared to the residual
Newtonian constant ansatz by $\Delta \chi^2 = -15$. Similar improved fits, also
occur in about $10\%$ of Monte Carlo realizations of Newtonian data. Thus, the
significance level of this improved fit is at a level of not more than
$2\sigma$. The energy scale corresponding to this best fit wavelength is
identical to the dark energy length scale $\lambda_{de} \equiv\sqrt[4]{\hbar
c/\rho_{ de}}\approx 0.1mm$.
