Collatz Conjecture as a Problem of Diophantine Approximation: A New Parametrization and Research Program (RU and ENG)

This article introduces a new approach to the unsolved Collatz (3n+1) conjecture by reinterpreting it through the lens of Diophantine approximation. The core idea is to parametrize Collatz trajectories using a sequence of integers derived from the number of division steps, transforming the problem into one about the average behavior of this sequence. The main result establishes a precise equivalence: the Collatz conjecture is true if and only if, for any infinite trajectory generated by the process, the long-term average of this parameter always exceeds a specific constant. This provides a clear and novel criterion for proving or understanding the conjecture. The work identifies a fundamental obstacle termed the "two-scale problem," which explains the deep difficulty of the conjecture by revealing a conflict between established theorems in transcendental number theory and the structure of the Collatz equations. This analysis demonstrates why many common finite or modular methods are inherently insufficient. Finally, the article moves beyond criticism to propose a concrete research program, defining specific mathematical problems—the "Baker-Collatz Problem"—whose solutions would lead to a complete proof. The contribution is a paradigm shift, reframing the conjecture from an intractable mystery into a structured set of tractable research objectives.