Tolerant sequence

In number theory, the Embree–Trefethen constant is a threshold value labelled β*.

For a fixed positive number β, consider the recurrence relation

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

where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "−".

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

$\sigma (\beta )=\underset{n\to \mathrm{\infty }}{\lim }(|x_n{|}^{1/n})$

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.

We have

σ < 1 for 0 < β < β* = 0.70258 approximately,

so solutions to this recurrence decay exponentially as n→∞ with probability 1, and

σ > 1 for β* < β,

so they grow exponentially.

Regarding values of σ, we have:

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

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

References

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