Precise asymptotics with log-periodic term in an elementary optimization problem

Abstract

The function ${\inf }_{n}n{x}^{1/n}$ has the asymptotics $eu+e{d}^2(u)/(2u)+O(1/u^2)$ as $x\to \infty$, where $u=\log x$ and $d(u)$ is the distance from $u$ to the nearest integer. We generalize this observation. First, the curves $y=n{x}^{1/n}$ can be written parametrically as $\log x=nt$, $y=nt$. In general, let $(u_n(t),v_n(t))$ be a family of parametric curves with asymptotics $u_n=n{p}_1(t)+q_1(t)+r_1(t)/n+O(1/n^2)$ and $v_n=n{p}_2(t)+q_2(t)+r_2(t)/n+O(1/n^2)$. Suppose the function $p_1(t)/p_0(t)$ has a unique nondegenerate minimum in the parameter domain. It is shown that the asymptotics of their lower envelope $v(u)={\inf }_{n,t}{v}_n(t)$, where $u=u_n(t)$, has the asymptotics of the form $v(u)=a_{0}u+a_1+\Phi (u)/u+O(1/u^2)$, where $\Phi$ is an affinely transformed function $d^2(\cdot )$. Second, note that $n{x}^{1/n}$ is the minimum of the sum $t_1+t_2/t_1+\cdots +t_n/t_{n-1}$ subject to the constraint $t_n=x$. We consider a similar asymptotic problem for the sums $t_1+t_2/(t_1+1)+\cdots +t_n/(t_{n-1}+1)$. Let $F_n(x)$ is the minimum value of the $n$-term sum under the constraint $t_n=x$. Define $F(x)={\inf }_n{F}_n(x)$. We show that $F(x)=eu-A+e{d}^2(u+b)/(2u)+O(1/u^2)$ with certain numerical constants $A$ and $b$. We present alternative forms of this optimization problem, in particular, a ``least action'' formulation. Also we find the asymptotics $F_n^{(p)}(x)=e\log n-A(p)+O(1/\log n)$ for the function arising from the sums with denominators of the form $t_j+p$ with arbitrary $p>0$ and establish some facts about the function $A(p)$.

49 pp, 8 figures