An Elementary Approach to Some Analytic Asymptotics

Fredman and Knuth have treated certain recurrences, such as $M(0) = 1$ and$$M(n + 1) = \mathop {\min }\limits_{0 \leqslant k \leqslant n} (\alpha M(k) + \beta M(n - k)),$$ where $\min (\alpha ,\beta ) \u3e 1$, by means of auxiliary recurrences such as h(x) = \left\{ {\begin{array}{*{20}c} {0\qquad {\text{if}}0 \leqslant x \u3c 1,} \\ {1 + h({x / \alpha }) + h({x / \beta }){\text{ if}}1 \leq x \u3c \infty .} \\ \end{array} } \right. The asymptotic behavior of $h(x)$ as $x \to \infty$ with $\alpha$ and $\beta$ fixed depends on whether ${{\log \alpha } / {\log \alpha }}$ is rational or irrational. The solution of Fredman and Knuth used analytic methods in both cases, and used in particular the Wiener–Ikehara Tauberian theorem in the irrational case. The author shows that a more explicit solution to these recurrences can be obtained by entirely elementary methods, based on a geometric interpretation of $h(x)$ as a sum of binomial coefficients over a triangular subregion of Pascal’s triangle. Apart from Stirling\u27s formula, in the irrational case only the Kronecker–Weyl theorem (which can itself be proved by elementary methods) is needed, to the effect that if is irrational, the fractional parts of the sequence $\vartheta ,2\vartheta ,3\vartheta , \cdots$, are uniformly distributed in the unit interval

On the asymptotic behavior of some Algorithms

A simple approach is presented to study the asymptotic behavior of some algorithms with an underlying tree structure. It is shown that some asymptotic oscillating behaviors can be precisely analyzed without resorting to complex analysis techniques as it is usually done in this context. A new explicit representation of periodic functions involved is obtained at the same time.Comment: November 200

Enumeration of walks reaching a line

We enumerate walks in the plane $\mathbb{R}^2$, with steps East and North, that stop as soon as they reach a given line; these walks are counted according to the distance of the line to the origin, and we study the asymptotic behavior when the line has a fixed slope and moves away from the origin. When the line has a rational slope, we study a more general class of walks, and give exact as well as asymptotic enumerative results; for this, we define a nice bijection from our walks to words of a rational language. For a general slope, asymptotic results are obtained; in this case, the method employed leads us to find asymptotic results for a wider class of walks in $\mathbb{R}^m$

Analysis of a Recurrence Arising from a Construction for Nonblocking Networks

Define f on the integers n \u3e 1 by the recurrence f(n) = min( n, minm|n( 2f(m) + 3f(n/m) ). The function f has f(n) = n as its upper envelope, attained for all prime n. The goal of this paper is to determine the corresponding lower envelope. It is shown that this has the form f(n) ~ C(log n)1 + 1/γ for certain constants γ and C, in the sense that for any ε \u3e 0, the inequality f(n) ≤ (C + ε)(log n)1 + 1/γ holds for infinitely many n, while f(n) ≤ (C + ε)(log \,n )1 + 1/γ holds for only finitely many. In fact, γ = 0.7878... is the unique real solution of the equation 2 -γ + 3-γ = 1, and C = 1.5595... is given by the expression C = ( γ( 2 -γ log( 2γ) + 3-γ log( 3γ ) )1/γ / ( γ + 1)( 15-γ logγ + 1( 5/2 + 3-γ ) ∑5 ≤ k ≤ 7 logγ + 1((k + 1)/1) ∑8 ≤ k ≤ 15 logγ + 1 ((k + 1)/1) )1/γ ). This paper also considers the function f0 defined by replacing the integers n \u3e 1 with the reals n \u3e 1 in the above recurrence: f0(x) = min{ x,inf1 \u3c y \u3c x (2f0(x/y) + 3f0(x/y) }. The author shows that f0(x) ~ C0( log x )1 + 1/γ, where C0 = 1.5586... is given by C0 = 6e( 2-γ log 2-γ + 3-γ log 3-γ )1/γ ( γ /( γ + 1))1 + 1/γ and is smaller than C by a factor of 0.9994...

An Elementary Approach to some Analytic Asymptotics

Fredman and Knuth have treated certain recurrences, such as M(0) = 1 and M(n+1) = min 0≤k≤n (αM(k) + βM(n-k)), where min(α,β) \u3e 1. Their treatment depends on certain auxiliary recurrecnes, such as h(x) = 0, if 0 ≤ x \u3c 1; h(x) = 1 + h(x/α) + h(x/β), if 1 ≤ x \u3c ∞. The asymptotic behavior of h(x) as x→∞ with α and β fixed depends on whether log α/ log β is rational or irrational. The solution of Fredman and Knuth used analytic methods in both cases, and used in particular the Wiener-Ikehara Tauberian theorem in the irrational case. We show that a more explicit solutions to these recurrences can be obtained by entirely elementary methods, based on a geometric interpretation of h(x) as a sum of binomial coefficients over a triangular subregion of Pascal\u27s triangle. Apart from Stirling\u27s formula, we need in the irrational case only the Kronecker-Weyl theorem (which can itself be proved by elementary methods), to the effect that if ϑ is irrational, the fractional parts of the sequence ϑ, 2ϑ, 3ϑ, ... are uniformly distributed in the unit interval