Embree–Trefethen constant

In number theory, the Embree–Trefethen constant is a threshold value labelled β* ≈ 0.70258. For a fixed positive number β, consider the recurrence relation

[math]\displaystyle{ x_{n+1}=x_n \pm \beta x_{n-1} \, }[/math]

where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "−". This is a generalization of the random Fibonacci sequence to values of β ≠ 1.

It can be proven that for any choice of β, the limit

[math]\displaystyle{ \sigma(\beta) = \lim_{n \to \infty} (|x_n|^{1/n}) \, }[/math]

exists almost surely. In informal words, the sequence behaves exponentially with probability one, and σ(β) can be interpreted as its almost sure rate of exponential growth.

β* ≈ 0.70258 is defined as the threshold value for which

σ(β) < 1 for 0 < β < β*,

so solutions to this recurrence decay exponentially as n → ∞, and

σ(β) > 1 for β > β*,

so they grow exponentially. (In both cases, with probability 1.)

Regarding values of σ, we have:

• σ(1) = 1.13198824... (Viswanath's constant), and
• σ(β*) = 1 (by definition).

The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.

External links

• Weisstein, Eric W.. "Random Fibonacci Sequence". http://mathworld.wolfram.com/RandomFibonacciSequence.html.