Dense Egyptian fractions

Every 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 denominator. We show that for every positive rational there exist representations as Egyptian fractions whose largest denominator is at most and whose denominators form a positive proportion of the integers up to , for sufficiently large ; furthermore, the proportion is within a small factor of best possible.