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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 $C_d$.

This paper improves on our previous work in several ways:

• The main theorem now includes a true “if and only if” condition, in the style of the sharp FKN theorem on the symmetric group.
• In the monotone case, we provide a theorem in which the approximating function is a monotone DNF.
• We find the optimal value of $C_d$, proving matching upper and lower bounds.

The previous work applied in the more general $A$-valued setting. The proof in this work applies in that setting as well, but we only formulated it in the Boolean setting. The missing piece is the $A$-valued Kindler–Safra theorem, which follows in a black-box function from the usual Kindler–Safra theorem, as outlined in the previous version.