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

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

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:

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


  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 doi:10.1098/rspa.1999.0412.

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