A classical theorem of Maillet states that every nonconstant rational function with rational coefficients maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether there exists a transcendental entire function with the same property. We resolve this question in the negative. More generally, we prove that every real-analytic function on an interval with this property is the restriction of a rational function in $\mathbb{R}(x)$. The underlying local result is quantitative and uniform: there exists an absolute constant such that, for every nonrational real-analytic function $f$, every nonempty open subinterval of its domain contains a Liouville number whose image under $f$ has irrationality exponent bounded by that constant. The main ingredient is a two-height counting estimate for rational approximation to nonrational analytic graphs, with the source and target denominators treated independently. Its proof combines an adaptive determinant argument with uniform Wronskian sublevel estimates and Farey separation; a nested-interval construction then yields the local quantitative result.
We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact K\"ahler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincar\'e mass. Thus linearizability is not needed for the geometric ergodic theorem...
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density a...
We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman--Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corre...
In 1928 H. Cartan stated a conjecture about holomorphic curves in $\mathbb{P}^n$ parametrized by the unit disc, omitting $n+2$ hyperplanes in general position. He proved it for $n=2$. In 1996 the first-named author constructed counterexamples for all $n\geq 3$, and proposed a modified form of Cartan's conjecture which...
We study the dependence of the transfinite Schreier hierarchy on the choice of fundamental sequences for the countable limit ordinals. With each Schreier family we associate an interval endpoint function. We prove that the bounding number is equal to $\omega_1$ exactly when the fundamental sequences may be chosen so th...
Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\mid A(x),\qquad x\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process \emph{congruence jumping}. We establish a general reciprocal-polynomial rule governing such jumps, wi...
Max A. Alekseyev, D. I. Khomovsky· 1 citation
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