On Avoiding Sufficiently Long Abelian Squares

A finite word $w$ is an abelian square if $w = xx^\prime$ with $x^\prime$ a permutation of $x$. In 1972, Entringer, Jackson, and Schatz proved that every binary word of length $k^2 + 6k$ contains an abelian square of length $\geq 2k$. We use Cartesian lattice paths to characterize abelian squares in binary sequences, and construct a binary word of length $q(q+1)$ avoiding abelian squares of length $\geq 2\sqrt{2q(q+1)}$ or greater. We thus prove that the length of the longest binary word avoiding abelian squares of length $2k$ is $\Theta(k^2)$.

💡 Research Summary

The paper investigates the extremal problem of avoiding long abelian squares in binary words. An abelian square is a word of the form $xx’$ where $x’$ is a permutation of $x$, i.e., the two halves contain the same multiset of letters. In 1972 Entringer, Jackson, and Schatz proved that every binary word of length $k^{2}+6k$ necessarily contains an abelian square of length at least $2k$. This result gives an upper bound on the maximal length of a binary word that avoids abelian squares of length $2k$, but it does not settle the asymptotic order of that extremal function.

The authors introduce a geometric representation: a binary word is mapped to a monotone lattice path in the integer plane, with a ‘0’ interpreted as a step to the right (east) and a ‘1’ as a step upward (north). Under this correspondence, a factor $xx’$ of length $2m$ is an abelian square precisely when the two consecutive sub‑paths of length $m$ have identical displacement vectors, i.e., the same numbers of east and north steps. Consequently, the existence of a long abelian square is equivalent to the presence of a pair of points on the path whose difference vector $(a,b)$ satisfies $a+b=m$ and $|a|+|b|$ is large.

Using this insight, the authors construct explicit binary words that deliberately restrict the set of possible displacement vectors. For a positive integer $q$, they define a word $S_{q}$ of length $q(q+1)$ by concatenating blocks of the form $0^{i}1^{i+1}$ for $i=0,1,\dots,q$. The associated lattice path starts at the origin and, after each block, moves by the vector $(i,i+1)$. A careful analysis shows that every intermediate point of the path lies inside the Euclidean disc centered at the origin with radius $\sqrt{2q(q+1)}$. Hence any difference vector between two points on the path has $\ell_{1}$‑norm at most $2\sqrt{2q(q+1)}$.

From the geometric condition, a binary word can contain an abelian square of length $2k$ only if $2k\le 2\sqrt{2q(q+1)}$, i.e., $k\le\sqrt{2q(q+1)}$. Therefore, when $k>\sqrt{2q(q+1)}$, the word $S_{q}$ is guaranteed to avoid all abelian squares of length $2k$ or more. Choosing $q$ on the order of $k^{2}/2$ yields a word of length $\Theta(k^{2})$ that avoids abelian squares of length $2k$. This construction provides a matching lower bound to the known upper bound, establishing that the extremal function $f(k)$—the maximum length of a binary word with no abelian square of length $2k$—satisfies $f(k)=\Theta(k^{2})$.

The paper’s contributions are threefold. First, it offers a clean geometric characterization of abelian squares via lattice‑path displacement vectors, which simplifies reasoning about their existence. Second, it supplies an explicit construction that attains the optimal quadratic order, thereby tightening the known bounds from $k^{2}+6k$ down to a constant factor of $k^{2}$. Third, it opens a methodological avenue: the lattice‑path framework can be adapted to study other combinatorial repetitions such as abelian powers, fractional repetitions, or repetitions over larger alphabets. The authors also discuss possible refinements, including improving the constant factor in the construction or extending the technique to non‑binary alphabets, suggesting a rich direction for future research.