The arithmetic of continued fractions in the field of $p$-adic numbers

Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we develop the arithmetic of continued fractions over the field of $p$-adic numbers. In particular, we provide a complete methodology to compute the $p$-adic continued fraction of the Möbius transformation and the bilinear fractional transformation of $p$-adic numbers. These allow any standard arithmetic operation over $p$-adic numbers to be performed. In great contrast with real continued fractions, we prove that the knowledge of arbitrarily many partial quotients of the initial continued fractions is not always sufficient to recover some partial quotients of the transformations. However, we prove that the set of elements for which this is not possible has Haar measure zero in $\mathbb{Q}_p$.

💡 Research Summary

This paper develops a systematic arithmetic for continued fractions over the p‑adic field ℚₚ, focusing on the computation of continued‑fraction expansions of Möbius (linear fractional) and bilinear fractional transformations. The authors begin by recalling the classical theory of real continued fractions and Gosp­er’s algorithm for handling Möbius and bilinear transformations in ℝ. They then discuss the peculiarities of p‑adic continued fractions, noting that unlike the real case there is no unique “floor” function and several competing algorithms exist (Ruban, Browkin I, Schneider, and more recent variants). The paper adopts the Ruban and Browkin I algorithms as the primary frameworks because they guarantee finite expansions for all rational p‑adic numbers, even though a full p‑adic analogue of Lagrange’s theorem (periodicity for quadratic irrationals) remains open.

Section 3 establishes metric results for the three p‑adic continued‑fraction expansions considered. The main theorem shows that for µ‑almost all x ∈ ℚₚ (µ being the Haar measure) the p‑adic valuations of the partial quotients are unbounded below, and the exact distribution of these valuations is derived. Consequently, the set of numbers for which the valuation remains bounded has Haar measure zero.

Building on these metric foundations, Sections 4–6 analyze how elementary p‑adic arithmetic interacts with the partial quotients. Lemma 4.1–4.5 give precise conditions under which the p‑adic valuation of a sum, product, or quotient can be inferred from the valuations of the operands. These lemmas are crucial for tracking how information propagates through the transformations.

In Section 5 the authors treat the Möbius transformation γ = (xα + y)/(zα + t) with rational coefficients satisfying xt − yz ≠ 0. Theorem 22 provides necessary and sufficient conditions for determining the p‑adic “floor” ⌊γ⌋ solely from the coefficients and the first partial quotient a₀ of α. The key condition (3) – namely vₚ(xα) < vₚ(y) and vₚ(zα) < vₚ(t) – guarantees that the output condition is satisfied and that the first partial quotient of γ can be computed. When (3) holds, the computation succeeds if and only if the valuation of the current partial quotient aₙ does not exceed a constant k = vₚ(x) − vₚ(z). If (3) fails, an “input transformation” is performed: the matrix representing the transformation is multiplied on the left by the matrix associated with the first partial quotient of α, effectively reducing the problem to a new pair (α₁, γ₁). Lemma 24 shows that after a finite number of such input transformations, condition (3) will hold unless one encounters the exceptional situations vₚ(xα + y) ≥ vₚ(x) + 1 or vₚ(zα + t) ≥ vₚ(z) + 1. Corollary 30 and Proposition 31 prove that these exceptional inequalities cannot persist at every step unless α is a root of the numerator or denominator of the Möbius map. Thus, for almost all α the algorithm terminates after finitely many input steps. However, Example 27 constructs a pathological α for which the output condition never becomes true, demonstrating that, unlike the real case, an arbitrary number of partial quotients of α does not guarantee recovery of the partial quotients of γ. The authors then invoke the metric results of Section 3 to argue that the set of such pathological α has Haar measure zero.

Section 6 extends the analysis to the bilinear fractional transformation
γ = (xαβ + yα + zβ + t)/(eαβ + fα + gβ + h),
with eight rational coefficients forming a rank‑2 matrix. Theorem 37 gives necessary and sufficient conditions for the p‑adic floor of γ, analogous to the Möbius case but now involving both α and β. When a collection of valuation inequalities (e.g., vₚ(xβ) < vₚ(y), vₚ(zβ) < vₚ(t), etc.) hold, the output condition simplifies dramatically to a single inequality (5):
min{−vₚ(aₙ), −vₚ(bₙ)} ≥ u, where u = vₚ(e) − vₚ(x).
Thus, if the partial quotients of both α and β have sufficiently negative valuations, the first partial quotient of γ can be computed directly. The same input‑transformation machinery as in the Möbius case is employed, alternating between using a partial quotient of α and of β. The authors prove that, for µ‑almost all pairs (α, β), the required valuation condition is satisfied after finitely many input steps.

Section 7 discusses how the results adapt to the more recent p‑adic continued‑fraction algorithm introduced in