bibkey: johnc2026mo516232r2
authors: John C
year: 2026
title: "When does the zero-sum sign count satisfy a linear recurrence?, MathOverflow 516232 revision 2"
doi: null
url: https://mathoverflow.net/questions/516232/when-does-the-zero-sum-sign-count-s-nd-varepsilon-in-dn-sum-vare
claim: "For every positive integer sequence with a constant integer recurrence, all finite integer digit sets have C-finite zero-sum prefix counts iff the minimal characteristic polynomial has no unit-circle roots, simple expanding roots, and root-of-unity ratios between every pair of expanding roots."
strata_touched:
  - D5/S1/Recurrence/MoZeroSumCountRefutation
license: CC-BY-SA-4.0
triage: anchor

The zero-sum digit-count characterization

The source is John C’s MathOverflow question 516232, revision 2, revision GUID 583B6E92-78BD-4BC4-89F2-21EB26E02150, edited 2026-10-21 18:01:27 UTC. The pinned source text and its attribution are available in the released target record. No current web state is inferred from that versioned text.

For a sequence of positive integers satisfying a constant integer recurrence, the displayed characterization asserts that, for every finite integer set $D$, the exact counts $$S_n(D)=\#\{(d_1,\dots ,d_n)\in D^n:\sum _{i=1}^n{d}_i{a}_i=0\}$$ satisfy a constant-coefficient recurrence if and only if the minimal characteristic polynomial has no unit-circle roots, has only simple roots outside that circle, and has root-of-unity ratios between any two roots outside it. The source starts at $n=1$, places no monotonicity condition on the sequence, and bounds neither the digit alphabet nor recurrence order. It also describes the count property by rationality of its generating function.

The counterexample has $a_{2k-1}=4^k$, $a_{2k}=2^k$. The finite carry argument is a verification of this construction. Standard finite automata and bounded binary carries are not claimed as new methods.

The supplied scientific comparison checks the accessible Pisot-base zero-language statement attributed to Christiane Frougny’s “Representations of numbers and finite automata,” Mathematical Systems Theory 25 (1992), 37–60, DOI 10.1007/BF01368783, as restated in Frougny–Steiner, arXiv:0803.2874v3. That restatement supplies a standard single-base ingredient. It does not by itself give the original-coordinate exponent-band counting bridge for this alternating sequence. The original 1992 article has not been obtained in full; complete earlier coverage and historical priority remain UNKNOWN. The supplied comparison of the stated results in arXiv:2012.15135v3 concerns periodic representations and does not establish this counting equivalence. These bounded comparisons establish neither global novelty nor absence of a prior refutation.

Source attribution: John C, MathOverflow, revision identified above, CC BY-SA 4.0. The mathematical formulation here paraphrases that source; the exact source is the linked question and its pinned revision.

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https://mathoverflow.net/questions/516232/when-does-the-zero-sum-sign-count-s-nd-varepsilon-in-dn-sum-vare

Revision 2, GUID 583B6E92-78BD-4BC4-89F2-21EB26E02150; the literal statement is supplied by the pinned source record. This locator identifies the source and does not assert its present answer status.