Abelian powers in paper-folding words

We show that paper folding words contain arbitrarily large abelian powers.

💡 Research Summary

The paper proves that every infinite paper‑folding word contains abelian powers of arbitrary size. A paper‑folding word is generated from a binary instruction sequence b₀b₁b₂… (using the alphabet {1, −1}) by the rule f_i = (−1)^{j} b_k where i = 2^{k}(2j+1). The symbol −1 is called a “−1 of order k” when i satisfies a specific congruence modulo 2^{k+2}.

An abelian m‑power is a concatenation w₀w₁…w_{m−1} where each block w_i has the same multiset of letters, i.e. |w_i|a = |w_j|a for all letters a and all i, j. To detect such structures the authors introduce D{k,b}(ℓ,n), the number of order‑k −1’s in the interval (ℓ,n]. Lemma 2 shows that D{k,b}(ℓ,n) = ⌊(n−ℓ)/2^{k+2}⌋ + ε_{k,b}(ℓ,n), where ε∈{0,1} records whether the interval’s tail contains an extra −1. This decomposition is obtained by partitioning the interval into blocks of length 2^{k+2}, each of which contains exactly one index congruent to (2+b)·2^{k} (mod 2^{k+2}).

The central object is the vector Δ(s,d,m) ∈ ℕ^{m}. For each order k and sign b, define the binary vector
E_{k,b}(s,d,m) = (ε_{k,b}(s,s+d), ε_{k,b}(s+d,s+2d), …, ε_{k,b}(s+(m−1)d, s+md)).
Then Δ(s,d,m) = Σ_{k≥0} E_{k,b_k}(s,d,m). The i‑th coordinate of Δ measures the deviation of the actual number of −1’s in the i‑th block from the expected number based solely on length d. If Δ(s,d,m) is a constant vector (all coordinates equal), the corresponding factor f_{s+1}…f_{s+md} is an abelian m‑power.

Lemma 3 establishes an additivity property for Δ‑vectors: if s′ and d′ are even, r is an integer satisfying 2^{r} > s + m d, and for every i ≥ 0 the implication
E_{i,1}(s′,d′,m) ≠ E_{i,−1}(s′,d′,m) ⇒ b_i = b_i + r
holds, then
Δ(s,d,m) + Δ(s′,d′,m) = Δ(s + 2^{r}s′, d + 2^{r}d′, m).
The proof separates the cases k ≤ r−1 (where the congruence classes line up directly) and k ≥ r (where the extra shift 2^{r}d′ introduces a new −1 exactly when the condition on b_i is satisfied). This lemma allows one to “glue” smaller Δ‑vectors into larger ones by scaling and translation.

To obtain building blocks with simple Δ‑vectors, the authors construct intervals free of high‑order −1’s. Lemma 4 defines numbers ℓ_t (depending only on b_t,…,b_{t+3}) such that for the interval (ℓ_t, ℓ_t+2^{t}+2−1] we have D_{k,b}(ℓ_t, ℓ_t+2^{t}+2−1)=0 for all k ≥ t. Consequently, any sub‑interval of length 2^{t} contains no −1 of order ≥ t. Lemma 6 extends this to show that for any u ≤ t and any offset p, the vectors E_{k,b}(ℓ_t−1+2up, 2u, 2t−u) are zero for all k ≤ u−2 or k ≥ t−1 (and even for all k when b varies).

Lemma 7 uses the zero‑blocks from Lemma 6 to build a family of Δ‑vectors that sum to a constant vector. For fixed t ≥ 2 and 1 ≤ u < t, let D = Σ_{p=0}^{2^{t−u}−1} Δ(ℓ_t−1+2up, 2u, 2t−u). By expanding the definition of Δ and using the vanishing of E‑vectors outside the range u−1 ≤ k ≤ t−2, each coordinate of D simplifies to (t−2)·2^{t−u−1}, independent of the coordinate index. Hence D is a constant vector.

The main theorem (Theorem 8) combines these ingredients. Given any even m = 2q, choose t ≥ 2 and u ≥ 1 with t−u = q. Because the instruction sequence b is infinite, the finite pattern b_{u−1}b_{u}…b_{t+2} occurs infinitely often. Starting with the base Δ‑vector Δ(ℓ_t−1, 2u, m) (which is constant by Lemma 7), Lemma 3 is applied repeatedly. At step j we set
s_{j+1} = s_j + 2^{r_j}(ℓ_t−1 + 2u·j),
d_{j+1} = d_j + 2^{r_j}·2u,
where r_j is chosen large enough to satisfy the inequality 2^{r_j} > s_j + m d_j and the sign‑shift condition (5). Each iteration adds another copy of the constant Δ‑vector, so after m steps we obtain Δ(s_m, d_m, m) = m·Δ(ℓ_t−1, 2u, m), which is constant. Therefore the factor f_{s_m+1}…f_{s_m+md_m} is an abelian m‑power.

The paper concludes with explicit examples. For the regular paper‑folding word (all b_i = 1) and m = 4, the authors take u = 1, t = 3, compute ℓ_2 = 28, and exhibit the four Δ‑vectors Δ(28,2,4), Δ(30,2,4), Δ(32,2,4), Δ(34,2,4). By successive applications of Lemma 3 with scaling factors 2⁶, 2¹², and 2¹⁸, they obtain a final Δ‑vector (3,3,3,3), confirming a concrete abelian 4‑power of length 8. A more economical construction uses Δ(6,1,4) and Δ(0,2,4) to directly produce the constant vector (1,1,1,1).

In summary, the authors provide a rigorous, constructive proof that every paper‑folding word contains abelian powers of any prescribed size. The proof hinges on a fine‑grained modular analysis of the positions of −1’s, the introduction of the Δ‑vector encoding the deviation from expected counts, and a powerful additivity lemma that permits the assembly of large constant‑Δ blocks from smaller ones. This methodology not only resolves the question raised by Currie and Silva but also offers a versatile framework for studying abelian repetitions in other automatic or morphic sequences.