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Lemma 46 ([Product tests under the key reference]). For a fixed query shape and arbitrary global single-key functions f_ j,n with f_ j,n ≤1 , ∫∏_jf_ j,n (k_j)\,dν_n =∏_j∫f_ j,n \,dΩ_ l_j,q_j,n +O(2^ -N/2 ). The bound is uniform over the …

Lemma 46 ([Product tests under the key reference]). For a fixed query shape and arbitrary global single-key functions fj,n with fj,n ≤1 , ∫∏jf_ j,n (kj)\,dνn =∏j∫f_ j,n \,dΩlj,qj,n +O(2-N/2 ). The bound is uniform over the functions.

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证明
Lemma 25 ([Walsh estimates]). Let all weights below have absolute value at most…Lemma 45 ([Comparison of two bounded batches]). Consider two bounded batches of…观点
Start with Ω_n and expand the off-position Gram conditions. For any nonzero character, choose an interacting pair of positions and fix all other positions. The remaining phase has rank at least N between the two selected position groups. The prescribed single-position diagonal densities and the unary functions are separately bounded weights.Lemma 25 ([Walsh estimates]) gives the error above. The identical expansion with all functions one gives the normalization of the off-position Gram event. Its zero-character contribution is a fixed positive constant, and all other contributions are exponentially small. Dividing proves the assertion.
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