Doorn 2026 completeness exponentially increasing sequences

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corollary_1: If alpha is at least the golden ratio and some m >= 1 lies strictly between s_1 + ... + s_r and s_(r+1) for some r >= 0, then S_t(alpha) is not complete; the criterion behind the paper's non-completeness results.

lemma_1: If m is not a sum of distinct terms of S_t(alpha), lies strictly between s_1 + ... + s_r and s_(r+2), and s_n + s_(n+1) <= s_(n+2) for all n > r, then adding s_(r+3) + s_(r+5) + ... + s_(r+2k+1) to m gives a non-sum for every k >= 1; a lemma Graham attributes to Folkman.

lemma_4: For t >= 1 the terms of S_t(alpha) at most double from step to step: for all n >= 1 when 1 < alpha < 3/2, for n >= 2 when 3/2 <= alpha < phi, and for n >= 3 when phi <= alpha < 5^(1/3).

lemma_5: For t >= 1 and 1 < alpha < phi, if every m in [X, X + s_(r+1)) is a sum of distinct elements of {s_1, ..., s_r} for some positive integers r and X, then S_t(alpha) is complete; the certificate behind Propositions 7 to 9.

proposition_1: If alpha is not in [1, 2], S_t(alpha) is not complete for any t > 0; if alpha = 1, it is (entirely) complete if and only if t lies in [1, 2).

proposition_2: At alpha = 2 the sequence S_t(alpha), indexed from n = 1, is (entirely) complete if and only if t = 1/2^k for some k >= 1.

proposition_3: For 5^(1/3) <= alpha < 2 there is no t >= 1 for which S_t(alpha) is complete.

proposition_4: For phi <= alpha < 5^(1/3), S_t(alpha) is (entirely) complete if and only if t < min(3/alpha^2, 5/alpha^3).

proposition_5: For 3/2 <= alpha < phi, S_t(alpha) is entirely complete if and only if t < 3/alpha^2; in particular it is entirely complete for all these alpha when t <= (9 - 3 sqrt 5)/2.

proposition_6: For 1 < alpha < 3/2, S_t(alpha) is entirely complete if and only if t < 2/alpha; in particular it is entirely complete for all these alpha when t <= 4/3.

proposition_7: If 1 < alpha <= 5/4, then S_t(alpha) is complete for all t < 4/alpha; the paper's hand-checked example of a finite certificate below phi.

proposition_8: S_t(alpha) is complete for all t <= 3 when 1.3 < alpha <= 1.4, all t <= 5 when 1.2 < alpha <= 1.3, all t <= 10 when 1.1 < alpha <= 1.2, and all t <= 50 when 1 < alpha <= 1.1; a computer-assisted result.

proposition_9: S_t(alpha) is complete for every alpha with 1 < alpha <= 1 + 1/(ceil(t) + 2 ceil(sqrt t)), a completeness region of infinite area in the (t, alpha) plane.

theorem_p1: For 1 < alpha <= 5^(1/3) the sequence of floors of t alpha^n is entirely complete exactly when t < min(2/alpha, 3/alpha^2, 5/alpha^3), and for alpha at least the golden ratio it is not complete once t >= max(min(3/alpha^2, 5/alpha^3), 1), which with Graham's results settles every alpha >= phi.

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Wouter van Doorn, Completeness of exponentially increasing sequences. arXiv preprint (2026). arXiv:2602.23394v1, 25 Feb 2026.

For S_t(alpha) with s_n = floor(t alpha^n), the paper pushes Graham's 1964 method well past his square 0 < t < 1, 1 < alpha < 2. It proves that for 1 < alpha <= 5^{1/3} the sequence is entirely complete if and only if t < min(2/alpha, 3/alpha^2, 5/alpha^3), and that for alpha at least the golden ratio phi with t >= max(min(3/alpha^2, 5/alpha^3), 1) the sequence is not complete, which combined with Graham's results finishes the case alpha >= phi. Propositions 1-6 give the case analysis (alpha outside [1, 2] and alpha = 1, alpha = 2, 5^{1/3} <= alpha < 2, phi <= alpha < 5^{1/3}, 3/2 <= alpha < phi, 1 < alpha < 3/2). Propositions 2-4 get non-completeness from Corollary 1, a criterion for alpha >= phi derived from a lemma Graham attributes to Folkman (Lemma 1) on gaps in the set of subset sums. For 1 < alpha < phi the paper calls completeness for all t > 0 plausible but open, and describes a finite search through Lemma 5 that, where it succeeds, certifies completeness on a bounded region t < T, alpha < phi - eps; the paper does not prove that the search succeeds on every such region. It gives a human-readable example (Proposition 7: complete for all t < 4/alpha when 1 < alpha <= 5/4) and a computer-assisted one (Proposition 8: complete for all t <= 3, 5, 10, 50 when alpha lies in (1.3, 1.4], (1.2, 1.3], (1.1, 1.2], (1, 1.1] respectively, with data on GitHub). Proposition 9 supplies the infinite-area partial result: S_t(alpha) is complete whenever 1 < alpha <= 1 + 1/(ceil(t) + 2 ceil(sqrt t)). These results bear on problem 349, which the preprint leaves open for 1 < alpha < phi outside the regions it settles.

Source: https://arxiv.org/abs/2602.23394. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2602.23394), every other right reserved.

Bears on. #349: for every alpha >= phi the paper, with Graham's 1964 results for t < 1, determines which t give a complete sequence (p. 1); for 1 < alpha < phi it determines entire completeness and proves completeness on the regions of Propositions 7-9, leaving the problem's question open for the other pairs with t >= min(2/alpha, 3/alpha^2).

Results. Labels and pages are those of v1.

• Unnumbered results (p. 1): entire completeness for 1 < alpha <= 5^(1/3), non-completeness from phi.
• Lemma 1 (p. 2): a missing subset sum propagates (a lemma Graham attributes to Folkman).
• Corollary 1 (p. 3): the non-completeness criterion for alpha >= phi.
• Lemma 4 (p. 4): s_(n+1) <= 2 s_n for t >= 1 below 5^(1/3).
• Lemma 5 (p. 4): a run of s_(r+1) consecutive subset sums forces completeness.
• Proposition 1 (p. 4): alpha outside [1, 2], and alpha = 1.
• Proposition 2 (p. 4): alpha = 2.
• Proposition 3 (p. 5): 5^(1/3) <= alpha < 2 with t >= 1.
• Proposition 4 (p. 5): phi <= alpha < 5^(1/3).
• Proposition 5 (p. 5): entire completeness for 3/2 <= alpha < phi.
• Proposition 6 (p. 6): entire completeness for 1 < alpha < 3/2.
• Proposition 7 (p. 6): completeness for t < 4/alpha when 1 < alpha <= 5/4.
• Proposition 8 (p. 9): computer-assisted completeness boxes for 1 < alpha <= 1.4.
• Proposition 9 (p. 10): completeness for 1 < alpha <= 1 + 1/(ceil(t) + 2 ceil(sqrt t)).

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