Problem 349

../

claims/: The 6 claim pages of Problem 349, one per claimant's result; the problem's standing derives from them.

---

Statement. For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete (that is, all sufficiently large integers are the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$)?

Statement (corrected). For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$, $n\ge1$, complete (that is, all sufficiently large integers are of the form $\sum_{n\ge1}\varepsilon_n\lfloor t\alpha^n\rfloor$ with $\varepsilon_n=0$ or $1$ and $\sum_{n\ge1}\varepsilon_n<\infty$)?

Notes. The site's parenthesis asks for sums of distinct integers of the form $\lfloor t\alpha^n\rfloor$, so a value taken at several indices can be used only once. That changes the answer. At $t=1$, $\alpha=1$ every term is $1$: under the site's wording the only sums are $0$ and $1$, so the sequence is not complete, while with each term usable once every positive integer is a sum; the same holds for every $1\le t<2$ at $\alpha=1$. Below $\alpha=2$ coincident values matter as well: at $t=2/3$, $\alpha=1.7$ the terms begin $s_1=s_2=1$, $s_3=3$, and Graham's Theorem 2 makes the sequence entirely complete only when both ones are available, as Wouter van Doorn pointed out in the site's thread on 2025-09-07. The site's wording also names no first index, which changes the answer too: with the index from $n=0$ the pair $(1,2)$ gives $1,2,4,\ldots$ and is complete, and with the index from $n=1$ it gives $2,4,8,\ldots$ and is not.

The change replaces "the sum of distinct integers of the form $\lfloor t\alpha^n\rfloor$" by "of the form $\sum_{n\ge1}\varepsilon_n\lfloor t\alpha^n\rfloor$ with $\varepsilon_n=0$ or $1$ and $\sum_{n\ge1}\varepsilon_n<\infty$" and inserts "$n\ge1$" after the sequence; nothing else changes. The evidence is the posers' own text, the passage the site cites. Erdős and Graham [ErGr80] (Old and new problems and results in combinatorial number theory), printed p. 57, ask: "Let $S(t,\alpha)=(s_1,s_2,\ldots)$ with $s_n=[t\alpha^n]$. For what values of $t$ and $\alpha$ is $S(t,\alpha)$ complete?" Their chapter on completeness defines, printed p. 53, $P(A)=\{\sum_{k=1}^\infty\varepsilon_k\alpha_k:\varepsilon_k=0$ or $1,\ \sum_{k=1}^\infty\varepsilon_k<\infty\}$ for a sequence $A=(\alpha_1,\alpha_2,\ldots)$, notes on printed p. 54 that these sums "are restricted by the multiplicity any particular term can have", and calls a sequence $S=(s_1,s_2,\ldots)$ of integers complete "if $P(S)$ contains all sufficiently large integers" (printed p. 54). Graham's paper [Gr64e], §§1--2 (its card), states Erdős's conjecture for $S_t(a)=(s_1,s_2,\ldots)$, $s_n=[ta^n]$, with the same sums. No text of the posers counts values once or starts the index at $n=0$, so the defect is the site's. The monograph's card does not transcribe the printed p. 57 passage. The form follows the posers' statement of this question, not the results that settle parts of it, and the change moves no standing: the problem is open under the site's wording and under the corrected Statement.

Results about the site's wording are credited here and count for nothing. The Lean proofs contributed to formal-conjectures under the account cepadugato in June 2026 (pull requests #4225 and #4233, proofs pinned at 23c629bc and 19e39e33) state the site's wording, with values counted once and the index from $n=0$. Their non-completeness results for $0<\alpha\le1$ and for positive integer pairs other than $(1,2)$ answer only that wording; their completeness results at $\alpha=2$ carry over to the Statement, and their claim page records them with that scope.

Formulation. The site's wording read as written is a variant: sums of distinct values, the reading the formal-conjectures statement takes (the set of values $\lfloor t\alpha^n\rfloor$, $n\ge0$). Every sum of distinct values is a sum of distinct terms, so completeness under the variant implies completeness under the Statement, and the two differ only where several indices give the same value. At $\alpha=1$ the variant is never complete, while the Statement's sequence is complete exactly for $1\le t<2$; the variant is answered for $\alpha>2$, for $0<\alpha\le1$ and at integer pairs by the catalog's Lean results, and Kitamura's base $\sqrt\varphi$ and Geneson's even terms hold under it as well. An index from $n=0$ only renames the pairs: the pair $(t,\alpha)$ with the index from $n=0$ is the Statement's pair $(t/\alpha,\alpha)$, so the catalog's complete pairs $(1/2^k,2)$, $k\ge0$, are the Statement's $(1/2^{k+1},2)$. The claim pages of Graham, van Doorn and Sothanaphan use the Statement's sums and index.

Status. Open, the site's label (OPEN as of 2026-10-06). Six partial claims are recorded and none settles the question. Accepted on refereed evidence: [[problems/additive_bases/E0349/claims/1964_01_01_graham|Graham's 1964 determination of the complete pairs with $0<t<1$, $1<\alpha<2$]]. Pending: [[problems/additive_bases/E0349/claims/2025_09_08_van_doorn|van Doorn's 2026 preprint]], which with Graham's results decides every $\alpha\ge(1+\sqrt5)/2$, classifies entire completeness for $1<\alpha\le5^{1/3}$ and proves completeness regions below the golden ratio; [[problems/additive_bases/E0349/claims/2026_03_09_sothanaphan|Sothanaphan's note with GPT-5.2 Thinking]], sharpening van Doorn's infinite-area region; [[problems/additive_bases/E0349/claims/2026_06_10_cepadugato|the catalog's Lean proofs of the elementary regions of the site's wording]], of which the completeness of the pairs $(1/2^k,2)$, $k\ge1$, carries over to the Statement; [[problems/additive_bases/E0349/claims/2026_09_05_kitamura|Kitamura's Lean theorem at the square root of the golden ratio]], which makes the sequence complete for every $t>0$ at that one base; and [[problems/additive_bases/E0349/claims/2026_09_06_geneson|Geneson's Salem-base counterexample]], a 2026 preprint stating that at one Salem number below the golden ratio there are arbitrarily large $t$ with every $\lfloor t\gamma^n\rfloor$ even, which refutes the completeness the site's remarks conjecture for every $t>0$ and $1<\alpha<(1+\sqrt5)/2$. The pairs with $1<\alpha<(1+\sqrt5)/2$ and $t\ge\min(2/\alpha,3/\alpha^2)$ are classified by no claim outside the regions and the base $\sqrt\varphi$ proved complete, so the derived standing is open/none.

Source. erdosproblems.com/349, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #349, https://www.erdosproblems.com/349.

References.

• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève, 1980; printed pp. 53, 54 and 57. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Gr64e] Graham, R. L., On a conjecture of Erdős in additive number theory. Acta Arith. 10 (1964/65), 63-70. Library home: graham_nd_conjecture_erdos_additive_number_theory.
• [vD26] van Doorn, W., Completeness of exponentially increasing sequences. arXiv:2602.23394 (v1 2026-02-25), 11 pages; unrefereed. Library home: doorn_2026_completeness_exponentially_increasing_sequences.
• [Ge26] Geneson, J., Deletion thresholds and exponential examples for complete sequences. arXiv:2609.25107 (v1 2026-09-20), 14 pages; Theorem 9, p. 11; unrefereed. Library home: geneson_2026_deletion_thresholds_exponential_examples_complete_sequences.

Formalization. Statement in formal-conjectures, which states the site's wording: completeness of the set of values $\lfloor t\alpha^n\rfloor$, $n\ge0$. It gives no evidence for the corrected Statement.

Current assessment

The site records Problem 349 as OPEN as of 2026-10-06, and the derived standing is open/none: every claim page is partial. The picture they give for the corrected Statement: the sequence is never complete for $\alpha>2$ or $\alpha<1$; at $\alpha=1$ it is complete exactly for $1\le t<2$; at $\alpha=2$ exactly for $t=1/2^k$ with $k\ge1$; for $(1+\sqrt5)/2\le\alpha<2$ the complete pairs are determined (Graham for $t<1$, van Doorn for $t\ge1$); for $1<\alpha<(1+\sqrt5)/2$ the sequence is entirely complete exactly when $t<\min(2/\alpha,3/\alpha^2)$, is complete on the further regions of van Doorn's Propositions 7--9 and Sothanaphan's note, is complete for every $t$ at $\alpha=\sqrt\varphi\approx1.2720$ (Kitamura's unreviewed Lean theorem), and is not complete at one Salem base near $1.25$ for arbitrarily large $t$ (Geneson), which refutes the conjecture of completeness for every $t>0$ below the golden ratio while leaving the classification there open. Only Graham's paper is refereed; the 2026 results are preprints, a shared note and merged catalog statements with third-party Lean proofs, none reviewed or accepted by the site.

Search. The site's thread and proof-claims tab, the formal-conjectures file and the linked library cards; no independent assessment of proof coverage.

Known Results

• Graham 1964 [Gr64e] (refereed): the complete pairs with $0<t<1$, $1<\alpha<2$ are determined, a region of area about $0.85$, refuting Erdős's conjecture of completeness for all such pairs; entire completeness for $t<1$, $1<\alpha\le5^{1/3}$; on that square complete if and only if entirely complete; for every $k$ some $t_k\in(0,1)$ whose set of complete bases has at least $k$ components (the site's remark); claim page.
• van Doorn 2026 [vD26] (preprint; announced in the thread 2025-09-08): not complete for $\alpha\notin[1,2]$; at $\alpha=1$ complete exactly for $1\le t<2$; at $\alpha=2$ exactly for $t=1/2^k$ with $k\ge1$; for $(1+\sqrt5)/2\le\alpha<5^{1/3}$ complete exactly when $t<\min(3/\alpha^2,5/\alpha^3)$, and for $5^{1/3}\le\alpha<2$ never when $t\ge1$, which with Graham decides every $\alpha\ge(1+\sqrt5)/2$; for $1<\alpha\le5^{1/3}$ entirely complete exactly when $t<\min(2/\alpha,3/\alpha^2,5/\alpha^3)$; complete for $t<4/\alpha$ when $1<\alpha\le5/4$, for $t\le3$, $5$, $10$, $50$ on $(1.3,1.4]$, $(1.2,1.3]$, $(1.1,1.2]$, $(1,1.1]$ (computer-assisted), and whenever $1<\alpha\le1+1/(\lceil t\rceil+2\lceil\sqrt t\rceil)$, a region of infinite area; claim page.
• Sothanaphan 2026, with GPT-5.2 Thinking (note shared in the thread, 2026-03-09; not held): the infinite-area region sharpened by a bounded amount in the denominator and further rectangles certified complete, one row already known; van Doorn called the gain marginal and thanked the poster, saying that van Doorn's own computations seemed to have been independently verified; claim page.
• Catalog partial results, June 2026 (Lean proofs in a fork of formal-conjectures under the account cepadugato, generated with Claude Code by the pull requests' own footer), stated for the site's wording with values counted once and the index from $n=0$: never complete for $\alpha>2$ or $0<\alpha\le1$; $(1,2)$ and $(1/2^k,2)$ complete; positive integer pairs complete only for $(1,2)$. Only the completeness at $\alpha=2$ carries over to the Statement, as the completeness of $(1/2^k,2)$, $k\ge1$; claim page.
• Kitamura 2026 (Lean development published on GitHub 2026-09-05 and announced in the thread of Problem 354; developed with ChatGPT and OpenAI Codex, using GPT-6 (Astra), by its README; not reviewed, not built here): at $\alpha=\sqrt\varphi\approx1.2720$ the sequence $\lfloor t\alpha^n\rfloor$ is complete for every $t>0$, under the Statement and under the site's wording with either index start; claim page.
• Geneson 2026 [Ge26] (preprint arXiv:2609.25107, 2026-09-20; its earlier ResearchGate note was submitted to the site's proof-claims thread on 2026-09-06), Theorem 9: through a theorem of Dubickas on fractional parts of powers of Salem numbers, at the Salem number $\gamma\approx1.25$ with minimal polynomial $x^{18}-x^{12}-x^{11}-x^{10}-x^9-x^8-x^7-x^6+1$ there are arbitrarily large $t$ with every $\lfloor t\gamma^n\rfloor$ even, so the sequence is not complete. That refutes the conjecture, recorded in the site's remarks, that the sequence is complete for every $t>0$ and every $1<\alpha<(1+\sqrt5)/2$, and says nothing about other bases or about a classification; the author discloses machine assistance; claim page.
• Open: the pairs with $1<\alpha<(1+\sqrt5)/2$ and $t\ge\min(2/\alpha,3/\alpha^2)$ outside the proved regions and the base $\sqrt\varphi$; whether $\lfloor(3/2)^n\rfloor$ is even, or odd, infinitely often (the site's remark on the difficulty).

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• doorn_2026_completeness_exponentially_increasing_sequences
• doorn_2026_completeness_exponentially_increasing_sequences / corollary_1
• doorn_2026_completeness_exponentially_increasing_sequences / lemma_1
• doorn_2026_completeness_exponentially_increasing_sequences / lemma_4
• doorn_2026_completeness_exponentially_increasing_sequences / lemma_5
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_1
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_2
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_3
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_4
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_5
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_6
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_7
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_8
• doorn_2026_completeness_exponentially_increasing_sequences / proposition_9
• doorn_2026_completeness_exponentially_increasing_sequences / theorem_p1
• geneson_2026_deletion_thresholds_exponential_examples_complete_sequences
• geneson_2026_deletion_thresholds_exponential_examples_complete_sequences / theorem_9
• graham_nd_conjecture_erdos_additive_number_theory