Infinite Words Avoiding Equal-Length, Equal-Sum Factors

Definitions and mathematical statement

Let $A\subset{\mathbb{Z}}$ be a finite nonempty alphabet. An infinite word over $A$ is a sequence $w=(a_n)_{n\geq 0}$ with $a_n\in A$. A factor is a finite contiguous block of this sequence. For a finite word $x=b_1\cdots b_m$, let $|x|=m$ and $S(x)=\sum_{j=1}^{m}b_j$. An additive square is a factor $xx'$ with $|x|=|x'|\geq 1$ and $S(x)=S(x')$.

Does there exist a finite nonempty set $A\subset{\mathbb{Z}}$ and an infinite word $w=(a_n)_{n\geq0}$ over $A$ containing no additive square? Explicitly, is it true that \[ \exists A\subset{\mathbb{Z}},\quad 0<|A|<\infty,\quad \exists (a_n)_{n\geq0}\in A^{\{0,1,2,\ldots\}},\quad \forall i\geq0\ \forall m\geq1,\qquad \sum_{j=0}^{m-1}a_{i+j}\ne\sum_{j=0}^{m-1}a_{i+m+j}, \] where $i$ and $m$ are integers?

The formulation uses the standard convention that the repeated blocks have positive length; the supplied wording does not explicitly impose this condition. If empty blocks were allowed as forbidden factors, every word would contain the empty additive square, so this convention requires confirmation in mathematical review.

Short English statement

Does some finite set of integers support an infinite sequence in which every two consecutive nonempty blocks of the same length have different sums? The supplied record calls this the PVHH sum-square avoidance problem and labels it open; that status is supplied dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001561$, source problem code catalogue entry, title: Infinite Words Avoiding Equal-Length, Equal-Sum Factors. The record gives proposed year 2010 and names no proposer.
• Supplied source citation: Mini-Workshop: Combinatorics on Words (2011), Oberwolfach Reports, pp. 2195--2244, DOI 10.4171/owr/2010/37. https://doi.org/10.4171/owr/2010/37. The supplied verification field identifies the statement as Problem 47 in the official report and describes this registry entry as the canonical record for a duplicated extraction; its supplied statement-status label is corrected_verified. These verification claims have not been independently checked here.
• Supplied literature reference: Au, Robertson, and Shallit, https://arxiv.org/abs/0812.2466. No paper title or specific supporting passage is supplied.
• Supplied literature reference labeled original workshop report: https://doi.org/10.4171/owr/2010/37.
• Literature assessment checked at 2026-08-21: the supplied curation labels the PVHH sum-square avoidance problem unresolved for infinite words over finite subsets of the integers. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.