On Shallit's minimization problem

TL;DR

This paper rigorously analyzes Shallit's 1994 minimization problem, proving properties previously known only numerically, establishing the asymptotic constant, and providing a sharp remainder estimate using dynamical systems analysis.

Contribution

It offers a rigorous proof of the asymptotic behavior and the constant in Shallit's minimization problem, advancing understanding beyond numerical evidence.

Findings

Proved existence of the asymptotic constant.

Established a sharp remainder estimate.

Analyzed trajectories of a planar discrete dynamical system.

Abstract

We revisit J. Shallit's minimization problem from 1994 SIAM Review concerning a two-term asymptotics of the minimum of a certain rational sum involving variables and products of their reciprocals, the number of variables being the large parameter. Properties previously known numerically, most importantly, the existence of the constant in the asymptotics, are proved. We supply a sharp remainder estimate to the originally proposed asymptiotic formula. The proofs are based on the analysis of trajectories of a planar discrete dynamical system that determines the point of minimum.