Sparse juntas on the biased hypercube

We give a structure theorem for Boolean functions on the biased hypercube which are $\epsilon$-close to degree $d$ in $L_2$, showing that they are close to sparse juntas. Our structure theorem implies that such functions are $O(\epsilon^{C_d} + p)$-close to constant functions. We pinpoint the exact value of the constant $C_d$.