Relative finite optimization of arithmetic approximation distances
Loading original first page…
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 Σ(n≥1) |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−β²)⁻¹ times the complete optimum, where β=2D(2D+1)/[(R+1)(R+2)]<1. The latter is a Kantorovich application and gives a sufficient cutoff R=O(N²ε^(−1/4)) for relative 1+ε 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∼N log N; at R∼N^(1+α) its full-to-prefix ratio tends to 1/α. 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.
Read the full paper in the original PDF.
Edition files
- Native_arithmetic_optimization.pdf95.6 KiB · pdf
SHA-256
49b469a335b1c6f95687b2847fb2c8ffa54ec6515501c55d964ecde32e8cac81 - Native_arithmetic_optimization.zip238.1 KiB · package
SHA-256
e048a66a3f0c530c0645925cc6d8b663f46e3b3c8285582c4f0559ae4b3efe3f
Available editions
Research details
- Subjects
- Mathematics · Number Theory · Numerical Analysis
- Keywords
- Nyman–Beurling criterion, Báez-Duarte distance, arithmetic approximation, complete tail, Kantorovich inequality, exact rational optimization, fractional parts
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
In the complete fractional-part sequence norm, rational finite optimization gives a candidate-quality factor (1−β²)⁻¹ for β<1 and sufficient cutoff O(N²ε^(−1/4)); exact adjacent-wavelet laws obstruct uniformly bounded short-prefix extension.
Disclosures
AI assistants supported derivation, literature comparison, exposition and independent checks. The author is responsible for the mathematical statements and presentation.
Reproduction
The 8-page manuscript and complete main.tex accompany build.py, CITATION.cff, README.md, MANIFEST.sha256, verify.py and results.json in an 8-member package. README.md gives Tectonic and Python 3.9+ standard-library commands. The verifier reproduces 12,333 exact rational controls and native optimization examples at N=4,8,12, retaining separate optimum brackets, candidate-norm brackets and quality factors. Universal infinite-tail bounds are proved analytically from the attributed Montgomery–Vaughan inequality.