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:
“For the time of towns is tolled from the world by funereal chimes, but in nature the universal hours are counted by succeeding tribes of animals and plants, and by growth of joy on joy.”
—Ralph Waldo Emerson (1803–1882)
“We honor motherhood with glowing sentimentality, but we don’t rate it high on the scale of creative occupations.”
—Leontine Young (20th century)
“The fundamental steps of expansion that will open a person, over time, to the full flowering of his or her individuality are the same for both genders. But men and women are rarely in the same place struggling with the same questions at the same age.”
—Gail Sheehy (20th century)
“Man is the only animal that can remain on friendly terms with the victims he intends to eat until he eats them.”
—Samuel Butler (1835–1902)