arXiv subjects

Giuliano Romeo

Publications and source records attributed to Giuliano Romeo.

14 recordsLinked to original sources

arXiv2026

Real convergence and periodicity of $p$-adic continued fractions

Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic $p$--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in $\mathbb{R}$ is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in $\mathbb{Q}_p$. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's $p$-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the $p$-adic digits of a quadratic irrational, that holds for almost all $p$-adic numbers.

math.NT↗

arXiv2026

Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.

math.NT↗

arXiv2026

Escalations and criteria over real quadratic fields

The famous 15-Theorem and 290-Theorem fully characterise universal quadratic forms over $\mathbb{Q}$. Similar theorems exist for every totally real number field, but only over $\mathbb{Q}(\sqrt5)$ the criterion set is explicitly known. We study criterion sets both theoretically and computationally: We develop the method of escalation over number fields, thus providing a simple proof of finiteness of the criteria and, more importantly, a practical tool for computing them. We illustrate this by explicit computations for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$, obtaining a conjecture about the corresponding universality criteria; we also prove universality of many escalator lattices. Moreover, we develop the somewhat different theory of escalations for diagonal quadratic forms, obtaining the diagonal criterion set for $\mathbb{Q}(\sqrt5)$ and conjecturally for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$.

math.NT↗

arXiv2026

Universality criterion sets for quadratic forms over number fields

In analogy with the 290-Theorem of Bhargava-Hanke, a criterion set is a finite subset $C$ of the totally positive integers in a given totally real number field such that if a quadratic form represents all elements of $C$, then it necessarily represents all totally positive integers, i.e., is universal. We use a novel characterization of minimal criterion sets to show that they always exist and are unique, and that they must contain certain explicit elements. We also extend the uniqueness result to the more general setting of representations of a given subset of the integers.

math.NT↗

arXiv2026

Representing rational integers by generalized quadratic forms over quadratic fields

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.

math.NT↗

arXiv2025

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 $\mathbb{Q}_p$.

math.NT↗

arXiv2025

On the arithmetic of multidimensional continued fractions

The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the Möbius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms.

math.NT↗

arXiv2025

Transcendence criteria for multidimensional continued fractions

Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization of classical continued fractions, introduced by Jacobi, and there are many well-studied open problems related to them. In this paper, we establish transcendence criteria for multidimensional continued fractions. In particular, we show that some Liouville-type and quasi-periodic multidimensional continued fractions are transcendental. We also obtain an upper bound on the naive height of cubic irrationals arising from periodic multidimensional continued fractions and exploit it to prove the transcendence criteria in the quasi-periodic case.

math.NT↗

arXiv2023

On alternative definition of Lucas atoms and their $p$-adic valuations

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.

math.NT↗

arXiv2023

Continued fractions in the field of p-adic numbers

Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued fractions defined over the field of $p$--adic numbers $\mathbb{Q}_p$, which in the last years has recorded a considerable increase of interest and research activity. We start from the very first definitions up to the most recent developments and open problems.

math.NT↗

arXiv2023

A new algorithm for p-adic continued fractions

Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still not known if a $p$--adic continued fraction algorithm exists that shares a similar property. In this paper we modify and improve one of Browkin's algorithms. This algorithm is considered one of the best at the present time. Our new algorithm shows better properties of periodicity. We show for the square root of integers that if our algorithm produces a periodic expansion, then this periodic expansion will have pre-period one. It appears experimentally that our algorithm produces more periodic continued fractions for quadratic irrationals than Browkin's algorithm. Hence, it is closer to an algorithm to which an analogue of Lagrange's Theorem would apply.

math.NT↗

arXiv2022

Special subsets of addresses for blockchains using the secp256k1 curve

In 2020 Sala, Sogiorno and Taufer have been able to find the private keys of some Bitcoin addresses, thus being able to spend the cryptocurrency linked to them. This result was unexpected, since the recovery of non-trivial private keys for blockchain addresses is deemed to be an infeasible problem. In this paper we widen this analysis by mounting a similar attack to other small subsets of the set of private keys. We then apply it to other blockchains as well, examining Ethereum, Dogecoin, Litecoin, Dash, Zcash and Bitcoin Cash. In addition to the results, we also explain the techniques we have used to perform this exhaustive search for all the addresses that have ever appeared in these blockchains.

cs.CR↗

arXiv2022

Convergence conditions for $p$--adic continued fractions

Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in $\mathbb{R}$. In particular, an analogue of the Lagrange's Theorem is not yet proved for any attempt of generalizing continued fractions in $\mathbb{Q}_p$. Thus, it is worth to study the definition of new algorithms for $p$--adic continued fractions. The main condition that a new method needs to fulfill is the convergence in $\mathbb Q_p$ of the continued fractions. In this paper we study some convergence conditions for continued fractions in $\mathbb{Q}_p$. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.

math.NT↗

arXiv2022

On the periodicity of an algorithm for p-adic continued fractions

In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in $\mathbb{Q}_p$ that have a periodic expansion with period of length four, solving an open problem left by Browkin.

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