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
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 can be interpreted as its almost sure rate of exponential growth.
β* ≈ 0.70258 is defined as the threshold value for which
so solutions to this recurrence decay exponentially as n → ∞, and
so they grow exponentially.
Regarding values of σ, we have:
The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.