2025/12/12 by Kumar, Veekesh, Prasad, Gorekh
Mathematics · #11B37 #11J68 #11J87(Primary) #11R06(Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2512.11337
openalex publication_date 2025/12/12 · openalex created_date 2025/12/16 · openalex updated_date 2026/07/28
For a complex number x, \Vert x\Vert:=min\|x-m|:m∈ℤ\. Let k≥ 1 be an integer, and K be a number field. Let α1,…,αk be algebraic numbers with |αi|≥ 1 and let di denotes the degree of αi for 1≤ i≤ k. Set d=d1+⋯+dk. In this article, we show that if the inequality 0<\Vertλ1 qαn1+⋯+λk qαnk\Vert<\fracθnqd+ε has infinitely many solutions in (n, q,λ1,…,λk)∈ ℕ2× (K^×)k with absolute logarithmic Weil height of λi is small compared to n and some θ∈ (0,1), then, in particular, the tuple (λ1 qαn1,…, λk qαnk) is pseudo-Pisot, and at least one of αi is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over ℚ. The case q=1 was recently proved by Kulkarni, Mavraki, and Nguyen \citekul, which is a generalization of Mahler's question proved in \citecorv. As a consequence of our result, we obtain the following generalization of this question: let α>1 be an algebraic number with d=[ℚ(α):ℚ]. For a given ε>0, if the inequality 0lt;\Vertλqαn\Vertlt;\fracθnqd+ε has infinitely many solutions in the tuples (n,q,λ)∈ ℕ2× K^× with absolute logarithmic Weil height of λ is small compared to n and θ∈ (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.