From China to IMPA: Hao Cheng defends thesis in Fractal Geometry

This Friday (16), Hao Cheng defends his doctoral thesis presenting advances in the understanding of the fractal geometry of variations of the Lagrange spectrum, a central theme in the interface between Dynamical Systems, Number Theory and Fractal Geometry. The work ” On the fractal geometry of Lagrange spectrum variations”The paper, supervised by researcher Carlos Gustavo Moreira, extends classic results in the area and reveals unexpected behaviors in spectra little explored in the literature. The defense will be broadcast live at 10:30 am, on IMPA’s YouTube.

The doctoral student’s relationship with mathematics began at an early age. Since elementary school, the subject had already piqued his interest, and he had the support of his family – even though he was the only mathematician in the house. “This support was essential for me to get to this place and this point in my life,” he says. He completed his academic training at Nankai University in China, where he worked under Professor Ming Li. It was there that he acquired a solid grounding in Dynamic Systems, an area he would continue to develop during his doctorate.

IMPA entered Hao Cheng’s path when Moreira – who would later become his advisor – visited Nankai University. “Meeting him and learning more about the research carried out at IMPA in Rio de Janeiro made this my main goal for my doctoral studies,” he recalls.

Adapting to life in Rio de Janeiro has been gradual. Although he is still getting used to the local cuisine, the mathematician highlights the city’s positive impact on his academic routine. “The natural setting provides a unique and motivating atmosphere for intense academic work.”

Exploring new frontiers in Lagrange spectra

In general, the thesis aims to generalize classical results on the Lagrange spectrum – a fundamental set of number theory linked to Diophantine approximation – to new variants that have not yet been studied.

The paper investigates three dynamic spectra: Lagrange’s second spectrum (L₂), its alternative version (L₂*) and a spectrum associated with the approximation of complex numbers of the form 1/2 + iα by Gaussian rational numbers. These objects arise in problems of approximation by rational fractions and are deeply linked to dynamic systems and fractal structures.

Using dynamic descriptions of these spectra through symbolic shifts and horseshoes, the PhD student extends a theory of dimension developed by researchers Moreira, Davi Lima and Christian Villamil to a more general context, in which the function considered is not smooth, but the maximum of a finite number of differentiable functions. This technical advance allows us to show, for example, that the function describing the Hausdorff dimension of the truncated L₂ spectrum is continuous, as well as proving that different notions of dimension coincide in this case.

One of the most surprising results of the thesis appears in the study of the alternative L₂* spectrum. Unlike what happens in classical spectra, for L₂* Hao Cheng found an abrupt phase transition: below a certain value, its Hausdorff dimension is zero, but any small increase in this parameter causes the dimension to jump to the maximum possible value.

The thesis also addresses the Diophantine approximation of complex numbers, showing that the problem can be reduced to the approximation of real numbers and modeled as a dynamic system of the type skew-product. As a result, the paper obtains new information about best approximation constants in complex numbers, deepening results previously studied by A. Schmidt.

After his defense, Hao Cheng intends to apply his mathematical training outside the academic environment. “My immediate goal is to make the transition to the professional world,” he says. He plans to work in a technology company, exploiting the analytical and problem-solving skills developed during his doctorate.