[Paper Review] ON CERTAIN FUNCTIONS AND RELATED PROBLEMS

Symon Serbenyuk|arXiv (Cornell University)|Jan 1, 2019

Education, Innovation and Language Studies6 references3 citations

TL;DR

This paper introduces generalized shift operators for Cantor series expansions and applies them to generalize the Gauss-Kuzmin problem and the Salem function. It establishes a unique, bounded, and continuous solution for a system of functional equations involving these operators, with explicit formulas for the solution and its Lebesgue integral, extending classical results to variable-base expansions and non-standard functional systems.

ABSTRACT

The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and generalizations of the Gauss-Kuzmin problem.

Motivation & Objective

• To generalize the Gauss-Kuzmin problem using generalized shift operators in variable-base Cantor series expansions.
• To extend the theory of singular and nowhere differentiable functions by introducing new classes via functional equations.
• To analyze the continuity, differentiability, and integrability of solutions derived from generalized shift operators.
• To investigate the structure of discontinuities and the behavior of solutions under iterated generalized shifts.
• To lay the foundation for future study of functional equations in diverse numeral systems, including Luroth, Engel, and continued fractions.

Proposed method

• Introduces the generalized shift operator σm for Cantor series expansions with variable bases (qk), defined by skipping the m-th digit and reweighting the remaining terms.
• Defines the generalized shift operator σm(x) as a piecewise continuous mapping that recombines partial sums of the series after removing the m-th term.
• Establishes a system of functional equations involving σnk−1 ◦ ... ◦ σn1(x), with coefficients βαnk,nk and weights pαnk,nk satisfying constraints in (−1,1) summing to 1.
• Uses iterative substitution to derive the solution g(x) = βαn1 + ∑_{k=2}^∞ βαnk ∏_{j=1}^{k-1} pαnj, proving convergence via decay of the product term.
• Applies the Lebesgue integral formula ∫₀¹ g(x)dx = (β₁ + ... + β_{q−1})/(q−1) to compute the integral of the solution function.
• Analyzes continuity and differentiability: g is continuous at q-irrational points and discontinuous only on a countable set depending on the sequence (nk).

Experimental results

Research questions

• RQ1 What is the limiting distribution of iterated generalized shift operators in variable-base Cantor series expansions?
• RQ2 How do functional equations involving generalized shift operators generate singular or nowhere differentiable functions?
• RQ3 Under what conditions does the solution to the functional equation system remain bounded and continuous?
• RQ4 What is the Lebesgue integral of the solution function in the q-ary case, and how does it depend on the coefficients βi and pi?
• RQ5 How does the structure of discontinuities in the solution function depend on the choice of the sequence (nk)?

Key findings

• The system of functional equations has a unique bounded solution g(x) = βαn1 + ∑_{k=2}^∞ βαnk ∏_{j=1}^{k-1} pαnj in the class of functions defined on [0,1].
• The solution g is continuous at all q-irrational points, with discontinuities forming a countable, finite, or empty set depending on the sequence (nk).
• The Lebesgue integral of g over [0,1] is ∫₀¹ g(x)dx = (β₁ + β₂ + ... + β_{q−1}) / (q−1), providing a closed-form expression.
• The convergence of the infinite series defining g is guaranteed by the decay of the product ∏_{t=1}^k pαnt → 0 as k → ∞, due to |pαnt| < 1.
• The generalized shift operator σm is continuous on open cylinders and differentiable almost everywhere with derivative qm, except at Q-rational points when n = m.
• The composition identity σₙₖ₋₁ ◦ ... ◦ σₙ₁(x) = σₙₖ₊ₙ₋₁(x) holds under specific conditions, enabling recursive analysis of iterated operators.