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Statement

Setting (p. 357). $S$ is a finite subset of $\mathbb{R}^n$, and an $S$-walk is a sequence $\{z_i\}$ with $z_{i+1}-z_i\in S$ for all $i$.

Theorem 3 (p. 363, quoted). "If $S$ has exactly three elements, then every $S$-walk of length nine has three collinear vectors; in fact three equally spaced collinear vectors."

Sharpness (p. 363). The paper states that summing the sequence $i,j,i,k,i,j,i$ of the orthonormal unit vectors $i,j,k$ (defined on p. 360) gives an $S$-walk of length eight with no three collinear points. Here the length counts the walk's vectors: the seven steps give eight partial sums, starting from the zero vector (an observation of this page).

The paper presents Theorem 3 as showing that Theorem 2 cannot be sharpened past a point: some restriction on collinearity survives in three dimensions (pp. 357 and 363).

Source. Joseph L. Gerver and L. Thomas Ramsey, On certain sequences of lattice points, Pacific J. Math. 83 (1979), no. 2, 357--363, doi:10.2140/pjm.1979.83.357, as identified on the source card. Theorem 3, its proof and the example are on p. 363.

Read depth. Claims checked: the statement and the example were read clause by clause on the printed page. Brown's theorem, on which the proof rests, was not checked here. Nothing here is independently reviewed.

Proof pointer

P. 363. The proof cites T. C. Brown (Amer. Math. Monthly 78 (1971), 886--888, the paper's reference [1]): any sequence of length nine on three symbols contains two adjacent segments that are permutations of each other; the paper notes that Brown's theorem can be checked by a direct computation of about an hour. Reading the three elements of $S$ as the symbols, two adjacent segments of the step sequence that permute each other have equal sums, so the walk's points at the start, the junction and the end of the two segments are collinear and equally spaced. The printed proof does not say how the length of a walk matches the length of its symbol sequence; under the count used in the example, a walk of length nine has eight steps (an observation of this page).

Bears on

• Problem 193: the problem asks whether every infinite walk in $\mathbb{Z}^3$ with steps from a finite set must contain three collinear points. By Theorem 3 every infinite walk whose step set has exactly three elements does, so a walk answering the problem negatively needs a step set of another size (an observation of this page).