Engel Expansion - Growth Rate of The Expansion Terms

Growth Rate of The Expansion Terms

The coefficients ai of the Engel expansion typically exhibit exponential growth; more precisely, for almost all numbers in the interval (0,1], the limit exists and is equal to e. However, the subset of the interval for which this is not the case is still large enough that its Hausdorff dimension is one.

The same typical growth rate applies to the terms in expansion generated by the greedy algorithm for Egyptian fractions. However, the set of real numbers in the interval (0,1] whose Engel expansions coincide with their greedy expansions has measure zero, and Hausdorff dimension 1/2.

Read more about this topic:  Engel Expansion

Famous quotes containing the words growth, rate, expansion and/or terms:

    We already have the statistics for the future: the growth percentages of pollution, overpopulation, desertification. The future is already in place.
    Günther Grass (b. 1927)

    We honor motherhood with glowing sentimentality, but we don’t rate it high on the scale of creative occupations.
    Leontine Young (20th century)

    Artistic genius is an expansion of monkey imitativeness.
    W. Winwood Reade (1838–1875)

    Picture the prince, such as most of them are today: a man ignorant of the law, well-nigh an enemy to his people’s advantage, while intent on his personal convenience, a dedicated voluptuary, a hater of learning, freedom and truth, without a thought for the interests of his country, and measuring everything in terms of his own profit and desires.
    Desiderius Erasmus (c. 1466–1536)