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Lemma 9 ([Large-denominator estimate]). Fix l≥1 and a finite prime set P as above. At each prime choose either ρ_p=μ_ l,p or ρ_p=μ_ l,p ^ . Let g_v:X_ P → be arbitrary functions. If, in one distinguished slot v , every Fourier coefficient at a denominator at most R vanishes, where R≥1 , then |∫∏_ u∈V g_u(z_u)\, d_ p∈ P ρ_p| ≤R^ -γ \|g_v\|_2 ∏_ u≠v \|g_u\|_ 2(d-1) .