Loglogskalor

Loglogskalor is a term used in some theoretical discussions to describe a slow-growth scale that relies on the double logarithm, log log n, as the primary measure of growth. It is employed to classify functions whose asymptotic behavior is much slower than any power of a logarithm, yet non-constant, by expressing growth in terms of the double logarithm.

Definition and scope

Let L(n) = log log n with the logarithm base being immaterial for asymptotic purposes. A nondecreasing function f defined on large n is said to have loglogskalor growth of order α ≥ 0 if, for some constants c1, c2 > 0, f(n) = Θ((L(n))^α) as n → ∞. The parameter α determines the principal growth rate within this scale. In practice, researchers use the class LG(α) to denote functions that behave like (log log n)^α up to constant factors for large n.

Examples and intuition

- α = 0 corresponds to constant growth.

- α = 1 corresponds to behavior proportional to log log n.

- α = 2 corresponds to behavior proportional to (log log n)^2.

These representatives illustrate a hierarchy that sits between constant, slowly varying logarithmic terms, and higher-order polylogarithms, emphasizing extremely slow escalation as n grows large.

Properties and relations

The loglogskalor classes LG(α) form a nested family: LG(α) ⊆ LG(β) for 0 ≤ α ≤ β. For any fixed α, functions in LG(α) grow slower than any polylogarithmic term in log n, such as (log n)^β with β > 0, and much slower than iterated logarithms of higher order.

Applications

The concept is mainly used in asymptotic analysis and complexity discussions to describe and compare borderline growth rates in algorithms, data structures, and number-theoretic questions where double-logarithmic behavior appears or is invoked for illustrative purposes.

See also

Logarithm, Logarithmic growth, Iterated logarithm, Polylogarithm, Slow variation.

Notes

Loglogskalor is a descriptive tool in theoretical contexts and is not a standard, universally adopted term in all mathematical literature.