Relative finite optimization of arithmetic approximation distances

Abstract

We give finite rational approximations to the complete least-squares distance from the constant sequence to fractional-part sequences {n/k} in the norm sum over n >= 1 of |f(n)|^2/[n(n+1)]. The exact periodic mean is a positive gcd covariance form. Montgomery–Vaughan’s circular Hilbert inequality yields a coefficient-uniform complete-tail error with constant D=N(N−1). This gives rational lower and upper optimization problems and a midpoint optimizer whose actual squared error is at most (1−beta^2)^(-1) times the complete optimum, where beta=2D(2D+1)/[(R+1)(R+2)]<1. The latter is a Kantorovich application and gives a sufficient cutoff R=O(N^2 epsilon^(-1/4)) for relative 1+epsilon quality of the optimizer. The computed optimum-value brackets are separate and retain their explicitly stated widths. An adjacent arithmetic wavelet shows that no bounded uniform prefix extension is possible at R asymptotic to N log N; at R asymptotic to N^(1+alpha) its full-to-prefix ratio tends to 1/alpha. The mean, Hilbert inequality, wavelet head identity and quadratic optimization tools are classical or previously established; their combination gives explicit native-norm truncation guarantees. A standard-library exact rational implementation is included.