  We present a study designed to measure the average LyC escape fraction
($\langle f_{\rm esc}\rangle$) of star-forming galaxies at z=3.5. We assemble a
sample of 148 galaxies from the VANDELS survey at $3.35\leq z_{\rm
spec}\leq3.95$, selected to minimize line-of-sight contamination of their
photometry. For this sample, we use ultra-deep, ground-based, $U-$band imaging
and HST $V-$band imaging to robustly measure the distribution of
$\mathcal{R_{\rm obs}}$ $=(L_{\rm LyC}/L_{\rm UV})_{\rm obs}$. We then model
the distribution as a function of $\langle f_{\rm esc}\rangle$, carefully
accounting for attenuation by dust, and the IGM (and CGM). A maximum likelihood
fit to the $\mathcal{R_{\rm obs}}$ distribution returns a best-fitting value of
$\langle f_{\rm esc}\rangle =0.07\pm0.02$, a result confirmed using an
alternative Bayesian inference technique (both exclude $\langle f_{\rm
esc}\rangle=0.0$ at $> 3\sigma$). By splitting our sample in two, we find
evidence that $\langle f_{\rm esc}\rangle$ is positively correlated with
Ly$\alpha$ equivalent width, with high and low sub-samples returning best fits
of $\langle f_{\rm esc}\rangle=0.12^{+0.06}_{-0.04}$ and $\langle f_{\rm esc}
\rangle=0.02^{+0.02}_{-0.01}$, respectively. In contrast, we find evidence that
$\langle f_{\rm esc}\rangle$ is anti-correlated with intrinsic UV luminosity
and UV dust attenuation; with low UV luminosity and dust attenuation
sub-samples returning best fits in the range $0.10 \leq \langle f_{\rm
esc}\rangle \leq 0.22$. We do not find evidence for a clear correlation between
$f_{\rm esc}$ and galaxy stellar mass, suggesting it is not a primary indicator
of leakage. Although larger samples are needed to further explore these trends,
they suggest that it is entirely plausible that the low dust and metallicity
galaxies found at z > 6 will display the $\langle f_{\rm esc}\rangle\geq0.1$
required to drive reionization.
