Sparse juntas on the biased hypercube

We give a structure theorem for Boolean functions on the p-biased hypercube which are ε-close to degree d in L₂, showing that they are close to sparse juntas. Our structure theorem implies that such functions are O(εCd + p)-close to constant functions. We pinpoint the exact value of the constant Cd. We also give an analogous result for monotone Boolean functions on the biased hypercube which are ε-close to degree d in L₂, showing that they are close to sparse DNFs. Our structure theorems are optimal in the following sense: for every d,ε,p, we identify a class Fd,ε,p of degree d sparse juntas which are O(ε)-close to Boolean (in the monotone case, width d sparse DNFs) such that a Boolean function on the p-biased hypercube is O(ε)-close to degree d in L₂ iff it is O(ε)-close to a function in Fd,ε,p.