Finite Beatty blocks and balanced gap words

Definitions and mathematical statement

R. Graham's source entry proposes studying weak $k$-term arithmetic progressions as a more flexible substitute for ordinary progressions, suggesting that standard problems might become easier. It does not state a particular extremal, density, Ramsey, or algorithmic conjecture. The stored delimiters appear as "b" and "c"; the supplied aid reads them as floor brackets. Under that explicit interpretation a weak progression is \[ \{\lfloor i\alpha+\beta\rfloor:1\le i\le k\}, \qquad \alpha\in{\mathbb{R}},\ \alpha\ge1,\quad\beta\in{\mathbb{R}}. \] The floor convention, rather than ceiling or nearest-integer rounding, matters for the strict endpoint conditions below. The parameter bound ensures distinct increasing terms.

The aid's classification problem takes an increasing integer sequence $a_1<\cdots<a_k$. It reports equivalence between being the displayed floor block and nonemptiness of \[ [1,\infty)\cap\bigcap_{1\le i<j\le k} \left(\frac{a_j-a_i-1}{j-i},\frac{a_j-a_i+1}{j-i}\right). \] For any $\alpha$ in this intersection, the parameter recovery is \[ \max_i(a_i-i\alpha)\le\beta<\min_i(a_i+1-i\alpha). \] The aid reports that these $O(k^2)$ exact rational inequalities construct admissible parameters. For $k=1$, the empty intersection places no additional restriction on $\alpha\ge1$, and the beta interval is nonempty.

Equivalently, its reported gap characterization is: there exists an integer $q\ge1$ such that all gaps $a_{i+1}-a_i$ lie in $\{q,q+1\}$, and the binary word \[ w_i=a_{i+1}-a_i-q\quad(1\le i<k) \] is balanced. Here balanced means that any two contiguous subwords of equal length have numbers of ones differing by at most one. For $k=1$, the gap condition is vacuous. The presence of two adjacent gap values alone is sufficient through $k=4$, but not generally.

The aid gives the sharp first failure at $k=5$: \[ \{0,1,2,4,6\},\qquad\text{gaps }1,1,2,2,\qquad w=0011. \] Its subwords $00$ and $11$ violate balance, and the strict slope intervals require both $\alpha<3/2$ and $\alpha>3/2$. The aid calls it a smallest counterexample to the gap-only criterion, meaning smallest number of terms, not a specified minimal diameter or lexicographic order. These are finite-block structural results; they do not settle any unspecified version of the standard additive-combinatorial problems mentioned programmatically by the source.

Short English statement

The source proposes replacing arithmetic progressions by finite floor-sequence blocks. The supplied aid characterizes such blocks using strict slope intervals and balanced adjacent-gap words, with a five-term example showing why two possible gap sizes alone are insufficient.

Sources

• Local UnsolvedMath registry, permanent ID $20001095$, source code catalogue entry; aim-combinatorics-notes.json, notes[219], tag problem.
• AIM workshop "Recent trends in additive combinatorics," Problem 1.14, R. Graham: https://aimath.org/WWN/additivecomb/additivecomb.pdf. Statement status exact, including the stored floor-delimiter artifacts. No PDF was accessed for this draft.
• No original remarks, literature field, proposer/year metadata, or authoritative external citation for the finite floor-block characterization is supplied beyond the in-text R. Graham attribution.
• Research attempt 1: context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; registry status open, classification OPEN-TRIAGE, suggested difficulty L2. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.