Embree–Trefethen constant

In number theory, the Embree–Trefethen constant is a threshold value labelled β* ≈ 0.70258.[1]

For a fixed positive number β, consider the recurrence relation

${\displaystyle x_{n+1}=x_{n}\pm \beta x_{n-1}\,}$

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

${\displaystyle \sigma (\beta )=\lim _{n\to \infty }(|x_{n}|^{1/n})\,}$

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.

References

1. Embree, M.; Trefethen, L. N. (1999). "Growth and decay of random Fibonacci sequences" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 455 (1987): 2471. Bibcode:1999RSPSA.455.2471T. CiteSeerX 10.1.1.33.1658. doi:10.1098/rspa.1999.0412.

External links

• Weisstein, Eric W. "Random Fibonacci Sequence". MathWorld.