Decomposition without L1 error and with near-optimal number of quadratic phases

Ascertain whether there exists a (possibly nonalgorithmic) decomposition theorem for 1-bounded functions f: F_2^n -> [-1, 1] that simultaneously achieves a number of quadratic phase functions r bounded by O(1/ε) and dispenses with any L1 error term, while controlling the Gowers U^3-norm error by at most ε.

Background

The authors give an efficient quadratic decomposition that writes a bounded function as a sum of at most O(1/ε) quadratic phases plus error terms with U^3-norm and L1-norm bounded by ε. They note that by using an alternative framework one can remove the L1 error term, but at the cost of increasing the number of quadratic phases to exp((1/ε)).

The unresolved question is whether one can obtain the “best of both worlds”: a decomposition that keeps the number of phases near-linear in 1/ε and simultaneously eliminates the L1 error term, even ignoring algorithmic efficiency. Resolving this would sharpen the structural understanding provided by quadratic decomposition theorems.