Graham 1964 finite sums reciprocals distinct nth powers

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corollary_1: A rational is a finite sum of reciprocals of distinct squares exactly when it lies in [0, pi^2/6 - 1) or in [1, pi^2/6).

corollary_2: A rational is a finite sum of reciprocals of distinct cubes exactly when it lies in one of four half-open intervals determined by zeta(3).

theorem_3: The set of reals approximable from above by finite sums of distinct reciprocal nth powers is a disjoint union of exactly 2^{t_n} half-open intervals, where t_n < (2^{1/n} - 1)^{-1} and t_n is asymptotic to n/ln 2.

theorem_4: Characterizes the rationals that are finite sums of reciprocals of distinct nth powers as a finite union of half-open intervals indexed by the subsums of the first t_n terms.

theorem_a: A rational p/q is a finite sum of distinct terms of the sequence of reciprocal nth powers exactly when, for every positive epsilon, some finite sum s of distinct terms satisfies 0 <= s - p/q < epsilon.

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Graham, R. L., On finite sums of reciprocals of distinct nth powers. Pacific J. Math. 14 (1964), no. 1, 85--92.

Graham characterizes the rationals representable as a finite sum of reciprocals of distinct nth powers of integers, for arbitrary fixed n. He starts from Theorem A (a consequence of his earlier work plus the fact that every sufficiently large integer is a sum of distinct nth powers): p/q lies in P(H^n) if and only if p/q is approximable from above-in-the-limit by finite subsums of H^n = (1^{-n}, 2^{-n}, 3^{-n}, ...). Theorem 4 then converts this into an explicit criterion: with t_n the largest k such that k^{-n} > sum_{j>=1} (k+j)^{-n} and P the set of subsums of the first t_n terms, p/q is such a finite sum if and only if p/q lies in the union over pi in P of the half-open intervals [pi, pi + sum_{k>=1} (t_n+k)^{-n}). Behind it is Theorem 3: Ac(H^n), the set of reals approximable from above by finite subsums, is that union, a disjoint union of exactly 2^{t_n} intervals, with t_n < (2^{1/n} - 1)^{-1} and t_n ~ n/ln 2. Corollaries 1 and 2 specialize the criterion to distinct squares and distinct cubes; for squares the stated consequence, which a footnote (p. 85) says Erdos also obtained, unpublished, is an explicit two-interval condition involving pi^2/6. Erdos problem 282 asks whether greedy algorithms with restricted denominators terminate; the site's commentary quotes Corollary 1's criterion for square denominators and asks whether the greedy algorithm for squares terminates. The paper supplies the existence criterion only and does not treat the greedy algorithm.

Source: https://msp.org/pjm/1964/14-1/p10.xhtml.

The copy read for this card is the publisher's file (Mathematical Sciences Publishers, 11 physical pages: a cover sheet, the printed pp. 85--92 as PDF pp. 2--9, an editors page and the volume contents); Pacific J. Math. 14 (1964), no. 1, 85--92, DOI 10.2140/pjm.1964.14.85 (Crossref record fetched), received May 13, 1963. The text layer garbles the displayed formulas, so the statements were read on the page images of pp. 85--92. Read status: claims checked. Theorem A, Definitions 1--3, Theorems 1--4, Corollaries 1 and 2 and Corollary A were read clause by clause; the proofs of Theorems 1--3 were read for structure only. No notice is printed on the cover sheet or the article pages; the publisher's article page shows "© Copyright 1964 Pacific Journal of Mathematics. All rights reserved." (https://msp.org/pjm/1964/14-1/p10.xhtml), every other right reserved.

Bears on. #282 (the site's [Gr64c]: Corollary 1 is the criterion x in [0, pi^2/6 - 1) or [1, pi^2/6) for sums of reciprocals of distinct squares that the commentary quotes, the case n = 2 of Theorem 4; these are existence criteria, and the paper says nothing about the greedy algorithm)

Results. Page numbers are the journal's (pp. 85--92).

• Theorem A (p. 85): p/q is a finite sum of distinct terms of H^n = (1^{-n}, 2^{-n}, ...) if and only if for every eps > 0 some finite sum s of distinct terms of H^n has 0 <= s - p/q < eps.
• Theorem 3 (p. 89): Ac(H^n) is the disjoint union of exactly 2^{t_n} half-open intervals [pi, pi + sum_{k>=1} (t_n+k)^{-n}), pi running over the subsums of the first t_n terms; t_n < (2^{1/n} - 1)^{-1} and t_n ~ n/ln 2.
• Theorem 4 (p. 91): the criterion above for finite sums of reciprocals of distinct nth powers.
• Corollary 1 (p. 91): distinct squares, p/q in [0, pi^2/6 - 1) or [1, pi^2/6).
• Corollary 2 (p. 91): distinct cubes, four intervals determined by zeta(3).

Corollaries A and B (p. 92), for distinct odd squares and for distinct squares congruent to 4 modulo 5, are stated as results that a more general form of Theorem A yields; they have no result pages.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.