19 Scalar Estimates for Buffered Filters
The conditional-entropy discount in a buffered rectangle of [ Ope26b , Lemma 3.2 ] optimizes positive filters on nested contours. This chapter collects two scalar ingredients: the weights of the contours and the iteration of the safe-box recurrence of [ Ope26b , Proposition 3.3 ] .
For positive costs \(c_j\) put \(Z=\sum _j c_j^{-1}\) and \(a_j=2/(Zc_j)\). Then \(\sum _ja_j=2\) and \(\sum _jc_ja_j^2=4/Z\), and every family of weights of total mass two has quadratic cost at least \(4/Z\). For \(c_j=2r+jD\), \(1\le j\le m\), with \(r\ge 1\), \(D{\gt}0\), \(P\ge 1\) and \(D(m+1)\ge (P-1)r\),
The mass and cost are direct computations, and optimality is Cauchy–Schwarz. For the lower bound, \(\log (x+D)-\log x\le D/x\) telescopes to \(\frac1D\log \frac{2r+D(m+1)}{2r+D}\), and \(2r+D(m+1)\ge (P+1)r\) while \(2r+D\le (2+D)r\).
Let \(F\) be nondecreasing on the positive integers, \(M\ge 2\) an integer, \(\lambda {\gt}M\) and \(K\ge 0\), with \(F(Mr)\le \lambda F(r)+Kr\) for all \(r\ge 1\). Then \(F(r)\le Cr^{\log _M\lambda }\) for all \(r\ge 1\). If \(M{\lt}\lambda {\lt}M^2\), then \(1{\lt}\log _M\lambda {\lt}2\).
By induction, \(F(M^j)\le \lambda ^jF(1)+K(\lambda ^j-M^j)/(\lambda -M)\le C\lambda ^j\). For \(M^{j-1}{\lt}r\le M^j\), monotonicity gives \(F(r)\le C\lambda ^j=C(M^j)^{\log _M\lambda }\le C(Mr)^{\log _M\lambda }\).