2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76807Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of the integers up to N, for sufficiently large N; furthermore, the proportion is within a small factor of best possible.16 pages; to appear in Trans. Amer. Math. SocNumber Theory11D68text