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On the p-adic Wirsing problem

This paper establishes a $p$ -adic counterpart to Poëls's improved lower bound for the Wirsing problem, proving that for any transcendental $p$ -adic number $\xi$ , the exponent $\omega_{n,p}^*(\xi)$ of approximation by algebraic numbers of degree at most $n$ satisfies $\omega_{n,p}^*(\xi)\geq\frac{n}{2-\log 2}-1$ .

Original authors: Anup B Dixit

Published 2026-08-19

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CC BY 4.0

Original authors: Anup B Dixit

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a persistent quest to understand how well we can approximate numbers that cannot be written as simple fractions. Some numbers, like the square root of two or the value of pi, are irrational; they go on forever without repeating. Even more elusive are transcendental numbers, which are not the solution to any simple polynomial equation with whole number coefficients. For over a century, mathematicians have been trying to measure how closely these mysterious numbers can be approached by algebraic numbers—numbers that are solutions to such equations. The challenge lies in balancing two competing factors: the complexity of the algebraic number being used for the approximation and the closeness of the fit. If the algebraic number is too simple, it cannot get very close; if it is very complex, it might get closer, but the rules of the game change. This field of study, known as Diophantine approximation, seeks to find the precise limits of this relationship. While the rules are well understood for ordinary real numbers, the situation becomes far more complicated when we shift our perspective to a different kind of number system used in advanced number theory, one where distance is measured not by how far apart numbers are on a line, but by how divisible their difference is by a specific prime number.

For decades, a famous result by the mathematician Wolfgang Wirsing provided a reliable floor for how well these approximations could work in the real number system. He proved that for any transcendental number, there are infinitely many algebraic numbers of a certain complexity that get close enough to satisfy a specific mathematical inequality. Recently, another researcher named Poëls significantly improved this floor, showing that the numbers can get even closer than Wirsing had predicted. However, this breakthrough was limited to the real number system. The question remained: does this improved limit hold true in the p-adic world, a parallel mathematical universe where numbers behave according to different rules of proximity? This is the central puzzle tackled by Anup B. Dixit in his recent paper. He set out to determine if the same high level of approximation is possible when we are working with these p-adic numbers, which are essential for understanding deep structures in number theory but are notoriously difficult to handle with the same tools used for real numbers.

Dixit's work confirms that the improved limit does indeed exist in the p-adic setting, though with a slight adjustment. He proved that for any transcendental p-adic number, there are infinitely many algebraic numbers that approximate it with a precision that matches the new, higher standard established by Poëls, minus a small, predictable penalty. To reach this conclusion, Dixit had to navigate a significant obstacle: the standard geometric tools used to find these approximations in the real world do not work in the p-adic world. In the real setting, mathematicians rely on a theorem about convex shapes to guarantee the existence of certain points, but no equivalent shape-based theorem exists for p-adic numbers. To solve this, Dixit invented a new method. Instead of trying to force the real-world tools to work, he constructed a special family of "congruence lattices." You can think of these as a structured grid of possibilities that organizes the polynomials in a way that mimics the behavior of a physical lattice, allowing him to apply classical geometric reasoning to a problem that previously seemed resistant to it.

The core of his strategy involved creating a large collection of polynomials that are small in value when evaluated at the target number. He then used a sophisticated algebraic technique involving generalized resultants—a way of combining multiple polynomials to create a new one—to isolate a single polynomial that satisfied very strict conditions. This new polynomial was then fed into a powerful tool called Hensel's lemma, which acts like a precise microscope in the p-adic world, allowing mathematicians to zoom in on a root of the polynomial and find the exact algebraic number that approximates the target. The result is a rigorous proof that the approximation limit is indeed higher than previously known for p-adic numbers. Specifically, the new lower bound for the approximation exponent is determined by a formula involving the degree of the algebraic numbers and a constant related to logarithms, which is slightly less than the best possible bound found in the real world.

This finding is significant because it closes a gap in our understanding of how numbers relate to one another across different mathematical systems. While the real-world result by Poëls was a major leap forward, Dixit's work ensures that this leap is not isolated to just one type of number system. By establishing a parallel result for p-adic numbers, he has shown that the fundamental limits of approximation are remarkably consistent, even when the rules of distance change. The slight reduction in the bound, which amounts to a loss of one unit in the exponent, is a direct consequence of the extra complexity introduced by the p-adic environment, specifically an additional factor that appears when estimating the size of certain determinants. This is not a failure of the method, but rather a precise accounting of the extra cost required to navigate the p-adic landscape. The paper stands as a testament to the power of adapting classical techniques to new environments, proving that even in the most abstract corners of mathematics, the search for order and precision continues to yield deeper insights into the nature of numbers.

Technical Summary: On the p-adic Wirsing Problem

Problem Statement
The paper addresses the $p$ -adic analogue of Wirsing's problem in Diophantine approximation. For a real transcendental number $\xi$ , the exponent $\omega^*_n(\xi)$ is defined as the supremum of real numbers $\omega$ such that there exist infinitely many real algebraic numbers $\alpha$ of degree at most $n$ satisfying $|\xi - \alpha| \leq H(\alpha)^{-\omega-1}$ , where $H(\alpha)$ is the height of the minimal polynomial of $\alpha$ . Wirsing established the lower bound $\omega^*_n(\xi) \geq \frac{n+1}{2}$ , a result recently improved by Poëls to $\omega^*_n(\xi) \geq \frac{n}{2} - \log 2$ using parametric geometry of numbers and generalized resultants.

The author investigates the $p$ -adic counterpart of this problem. Let $p$ be a prime and $\xi \in \mathbb{Q}_p$ be a transcendental number. The exponent $\omega^*_{n,p}(\xi)$ is defined as the supremum of $\omega$ for which there exist infinitely many algebraic numbers $\alpha \in \mathbb{Q}_p$ of degree $\leq n$ satisfying $|\xi - \alpha|_p \leq H(\alpha)^{-\omega-1}$ . Prior to this work, the best known lower bound in the $p$ -adic setting was the analogue of Wirsing's theorem, $\omega^*_{n,p}(\xi) \geq \frac{n+1}{2}$ , established by Morrison and Teulié.

Methodology and Strategy
The paper adapts the method developed by Poëls for the real case to the $p$ -adic setting. The core strategy involves:

Congruence Lattices: Since there is no direct $p$ -adic analogue of Minkowski's convex body theorem suitable for this specific construction, the author introduces a family of "congruence lattices" $\Lambda_h(\xi)$ . These lattices consist of integer polynomials of degree $\leq n$ whose values at $\xi$ are small in the $p$ -adic metric ( $|P(\xi)|_p \leq p^{-h}$ ). This recasts the $p$ -adic conditions into conditions on an ordinary lattice in $\mathbb{R}^{n+1}$ , allowing the application of classical geometry of numbers (specifically Minkowski's second theorem) to analyze successive minima.
Generalized Resultants: The author constructs a family of linearly independent polynomials derived from a "seed" irreducible polynomial and a shortest vector in the congruence lattice. Using a generalized resultant construction (similar to Poëls's), a basis of polynomials is formed.
Determinant Estimates and Hensel's Lemma: A determinant argument is employed at the $p$ -adic place. Unlike the real case, where the zeroth divided-Taylor coordinate estimate can sharpen the archimedean bound, the $p$ -adic estimate does not provide the same saving, leading to a slightly weaker bound. The author uses the strong form of Hensel's lemma to show that if a polynomial in the constructed basis has a sufficiently small value and derivative at $\xi$ , it possesses a root $\alpha \in \mathbb{Q}_p$ that serves as the required algebraic approximation.

Key Contributions and Results
The main result is the establishment of a new lower bound for the $p$ -adic approximation exponent:

Theorem 1.1: Let $p$ be a prime, $n \geq 2$ , and $\xi \in \mathbb{Q}_p$ be transcendental. Then:
$\omega^*_{n,p}(\xi) \geq \frac{n}{2} - \log 2 - 1$

This result improves upon the previous bound of $\frac{n+1}{2}$ for all $n \geq 6$ .

Significance and Limitations
The paper claims this result as a significant improvement in the $p$ -adic setting, bringing the lower bound closer to the real case established by Poëls. The author explicitly notes that the loss of the term "1" relative to Poëls's real result ( $\frac{n}{2} - \log 2$ ) reflects an additional height factor that arises in the $p$ -adic determinant estimate. Specifically, the archimedean determinant bound retains a factor of $H_1^{n-k+2}$ which cannot be sharpened in the $p$ -adic context as it can in the real context.

The paper does not claim to resolve the optimal bound (which is conjectured to be $n$ ) nor does it propose new applications beyond the theoretical advancement of the Wirsing problem in the $p$ -adic domain. The work is strictly confined to establishing this improved lower bound through the adaptation of Poëls's geometric and algebraic techniques to the $p$ -adic framework.

Technical Summary: On the p-adic Wirsing Problem

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