On the rational approximation to linear combinations of powers

Abstract: For a complex number $x$, $\Vert x\Vert:=\min{|x-m|:m\in\mathbb{Z}}$. Let $k\geq 1$ be an integer, and $K$ be a number field. Let $α_1,\ldots,α_k$ be algebraic numbers with $|α_i|\geq 1$ and let $d_i$ denotes the degree of $α_i$ for $1\leq i\leq k$. Set $d=d_1+\cdots+d_k$. In this article, we show that if the inequality $ 0<\Vertλ_1 qα^n_1+\cdots+λ_k qα^n_k\Vert<\frac{θ^n}{q^{d+\varepsilon}} $ has infinitely many solutions in $(n, q,λ_1,\ldots,λ_k)\in \mathbb{N}^2\times (K^\times)^k$ with absolute logarithmic Weil height of $λ_i$ is small compared to $n$ and some $θ\in (0,1)$, then, in particular, the tuple $(λ_1 qα^n_1,\ldots, λ_k qα^n_k)$ is pseudo-Pisot, and at least one of $α_i$ is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over $\overline{\mathbb{Q}}$. The case $q=1$ was recently proved by Kulkarni, Mavraki, and Nguyen \cite{kul}, which is a generalization of Mahler's question proved in \cite{corv}. As a consequence of our result, we obtain the following generalization of this question: let $α>1$ be an algebraic number with $d=[\mathbb{Q}(α):\mathbb{Q}]$. For a given $\varepsilon>0$, if the inequality $$ 0<\Vertλqα^n\Vert<\frac{θ^n}{q^{d+\varepsilon}} $$ has infinitely many solutions in the tuples $(n,q,λ)\in \mathbb{N}^2\times K^\times$ with absolute logarithmic Weil height of $λ$ is small compared to $n$ and $θ\in (0,1)$, then some power of $α$ is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.

• The paper establishes that exceptionally good rational approximations imply a pseudo-Pisot structure in power sum sequences.
• It employs Schmidt’s Subspace Theorem and height machinery to rigorously constrain algebraic integrality conditions and approximation rates.
• The results extend classical Diophantine approximation, offering new transcendence criteria for infinite products involving algebraic numbers.

Summary of "On the Rational Approximation to Linear Combinations of Powers" (2512.11337)

Introduction and Context

The paper investigates the Diophantine approximation properties of linear combinations of powers of algebraic numbers, focusing on the case where the absolute Weil height of the coefficients grows sublinearly with the power index. It generalizes key results from Mahler, Corvaja-Zannier, and Kulkarni-Mavraki-Nguyen, producing a Roth-type theorem for tuples $$\left(q\lambda_1\alpha_1^n + \ldots + q\lambda_k\alpha_k^n\right)$$ in considerable generality.

Given a sequence $(\alpha_1, \ldots, \alpha_k)$ of algebraic numbers of modulus at least $1$ and associated coefficients $\lambda_i$ of controlled (sublinear in $n$) logarithmic Weil height, the authors analyze under what circumstances the quantity $$\| \lambda_1 q\alpha_1^n + \ldots + \lambda_k q\alpha_k^n \|$$ can be approximated very closely by integers, as a function of $n$ and $q$. The level of approximation demanded refines earlier work, and the main theorems produce strong constraints on the algebraic structure of the $\alpha_i$.

Main Results

The core result establishes that if there exist infinitely many $(n, q, \lambda_1, \ldots, \lambda_k)$, with the Weil height of the $\lambda_i$ suitably bounded, such that

$$0 < \| \lambda_1 q\alpha_1^n + \cdots + \lambda_k q\alpha_k^n \| < \frac{\theta^n}{q^{d+\varepsilon}}$$

for some $\theta \in (0,1)$ and fixed $\varepsilon > 0$, then:

1. Pseudo-Pisot Nature of the Sequence: The sequence of tuples $$(\lambda_1 q\alpha_1^n, \ldots, \lambda_k q\alpha_k^n)$$ is pseudo-Pisot, that is, the Galois conjugates not appearing in the tuple itself all have modulus less than $1$ and the total trace of all conjugates is integral.
2. Algebraic Integrality: At least one $\alpha_i$ must be an algebraic integer, with all $\alpha_i$ algebraic integers if the combined Weil height of $q\lambda_i$ is controlled sublinearly.
3. Pisot-Type Characterization for $k=1$: In the single-term case, if the approximation is sharp for infinitely many $(n, q, \lambda)$ as above, then some power of $\alpha$ is Pisot.
4. Transcendence Applications: The authors rigorously deduce the transcendence of certain infinite products of algebraic numbers under mild separation and growth conditions.

The proofs rely on the advanced machinery of the Schmidt Subspace Theorem and effective height estimates, extending the rigidity phenomena observed for powers of algebraic numbers to much more general linear combinations.

Theoretical Implications

The results demonstrate a sharp rigidity: only in exceptional algebraic configurations—specifically, those associated with Pisot-type or pseudo-Pisot tuples—can heights remain comparable and approximation rates be as strong as the theorem prescribes. The characterization links rational approximation with the algebraic and arithmetic properties of the underlying numbers, tightly constraining the possible behavior when the approximation is "too good."

This provides an analogue of Roth’s theorem (and its many generalizations) for a substantially broader family of exponential-like sequences, and clarifies the precise arithmetic obstacles to "unexpectedly good" rational approximations in this setting.

Another theoretical implication is the generalization of the so-called “Mahler’s classification question” to sequences involving both variable coefficients and denominators.

Key Technical Tools

• Schmidt’s Subspace Theorem is crucially used to deduce that, unless the sequence is essentially “arithmetic” in nature (i.e., related to Pisot structures), "sufficiently good" approximations cannot be persistent.
• Height Machinery: Restriction to coefficients with sublinear logarithmic Weil height extensions is essential: if the heights are allowed to grow too fast, the result can fail.
• Galois Theoretic Reductions: Considerable structure is established for conjugates of the $\alpha_i$ and the action of Galois groups on the tuples under consideration.

Numerical and Structural Conclusions

The paper includes applications to transcendence theory. For example, the authors prove that if $\alpha > 1$ is a real algebraic number such that no power is Pisot, and if $a_n$, $b_n$ grow rapidly enough, then the infinite product

$$\prod_{n=1}^\infty \frac{[b_n \alpha^{a_n}]}{b_n \alpha^{a_n}}$$

is transcendental—greatly extending classical results and removing restrictive hypotheses on the Galois orbit’s modulus for $\alpha$.

Furthermore, the sharpness of the exponent in the denominator, $q^{d+\varepsilon}$, is shown via careful analysis of the approximation rate and the role of the degree $d$ of the number field.

Implications and Future Directions

The work clarifies that the obstacle to strong rational approximation for these sequences is not only found in powers of Pisot numbers but also in more elaborate algebraic configurations—the "pseudo-Pisot" tuples—highlighting the fine arithmetic structure underlying Diophantine approximation phenomena for linear recurrences and algebraic power sums.

Practically, this gives a new tool for distinguishing transcendental numbers arising from infinite products of the given form and offers criteria to test for transcendence in explicit examples, including those involving Salem numbers and other algebraic integers whose conjugates lie on or within the unit circle.

Future directions may include:

• Extending these criteria to inhomogeneous variants (with more general errors in approximation).
• Exploring uniformity and effective bounds for the number and nature of exceptions.
• Applying these techniques to non-linear recurrences or algebraic dynamical systems.

Conclusion

This article makes definitive progress in understanding how rational approximations to linear combinations of powers of algebraic numbers are governed by deep algebraic properties, specifically connecting the possibility of small fractional parts with the Pisot and pseudo-Pisot structure. In doing so, it generalizes and unifies a body of Diophantine approximation theorems and produces powerful applications to transcendental number theory, extending criteria and methods that were previously available only for much narrower classes of problems.