arXiv subjects

Maksim Lavrov

Publications and source records attributed to Maksim Lavrov.

2 recordsLinked to original sources

arXiv2026

Bounded Ratios of Best Approximations: Sharp Bounds and Spectra of Diophantine Exponents

We solve Problems 1, 2, and 3 posed by Marnat, Moshchevitin, and Schleischitz (arXiv:2505.15964). We consider totally irrational real n-by-m matrices and two conditions on their sequences of best approximations: (B) bounded successive height ratios and (C) bounded successive error ratios. Under (C), we prove that the ordinary Diophantine exponent is at most m and, when both dimensions are at least two, that the uniform exponent is at most m^2/(m+1). Combining these bounds with known results, we determine the individual spectra of both exponents under (C), with and without the additional assumption (B). When both dimensions are at least two, imposing (B) changes neither spectrum. A key ingredient is a block lemma for consecutive best approximations lying in a fixed rational subspace.

math.NT↗

arXiv2026

Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum

We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin $μ$ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound $μ\leq (448875+1611\sqrt{67773})/46982 = 18.4809\ldots$. We also exhibit an explicit gap $(L,R)$ in the Dirichlet spectrum, which yields the improved lower bound $μ\geq R = 5.9596\ldots$. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points $Z_k$, $k\geq 2$, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around $Z_k$ and describe the equivalence classes of all irrational numbers whose Dirichlet value is $Z_k$.

math.NT↗