Is it true that, for some k, if all (K-1)-words are encountered by a t-ary word at the same rate as a uniform random t-a

Definitions and mathematical statement

A $t$-ary word is a word over an alphabet $\Sigma$ with $|\Sigma|=t$. The question is explicitly about substitution instance counts: it asks whether there is a threshold $k$ such that, for every $K>k$, matching the uniform-random encounter rate for every $(K-1)$-word forces matching it for every $K$-word.

To expose the requested implication without supplying absent definitions, let $\mathcal P_m$ denote the source's intended class of $m$-words, let $I(P,W)$ denote its intended substitution-instance encounter rate of $P$ in $W$, and let $u_t(P)$ denote the corresponding rate for a uniform random $t$-ary word. These are placeholders for definitions missing from the record. The proposed threshold assertion has the form \[ \exists k\ \forall K>k\ \forall W,\qquad \left[\forall P\in\mathcal P_{K-1},\ I(P,W)=u_t(P)\right] \Longrightarrow \left[\forall P\in\mathcal P_K,\ I(P,W)=u_t(P)\right]. \] The quantifiers over $k,K$ are intended to concern word-size parameters, but their exact conventions, the class of host words $W$, and whether $k$ may depend on $t$ are not supplied. "Same rate" may be an exact limiting equality or an asymptotic condition for finite-word sequences; no normalization or limiting regime is given. The displayed schema therefore preserves the requested implication, rather than claiming a self-contained theorem. In particular, substitution-instance counts cannot silently be replaced by ordinary consecutive-substring frequencies.

Short English statement

Rorabaugh asks whether substitution-instance counts have a quasirandomness threshold: beyond some size, random-like counts for every pattern of one size would force random-like counts at the next size. The source does not define the counts or their limiting regime.

Sources

• Local UnsolvedMath registry, permanent ID 3100080, exact source code catalogue entry, attributed to Rorabaugh; status open, classification OPEN-TRIAGE.
• Joshua Cooper, Combinatorial Problems I Like, October 2020 snapshot, Words and Codes, source-order bullet 80, https://people.math.sc.edu/cooper/combprob.html. Supplied HTML extraction accessed July 29, 2026.
• The supplied research summary relates the question to Cooper's work on quasirandom words and substitution-instance counts, but provides no precise publication or defining reference. It says that a definitive resolution could not be verified. This is a supplied open/unverified assessment, not an independent current-status claim.
• Provenance flags status and rights NEEDS_REVIEW: the 2020 page labels the item open, later resolution was not checked item by item, and public access was verified without establishing redistribution permission.