Expositiones Mathematicae最新文献

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend and use it to prove that a number of vectors have an eventually periodic expansion in the Algebraic Jacobi–Perron algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive.

In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $\tau \left({\phi }^{\omega }\left(a\right)\right)$ where ${\phi }^{\omega }\left(a\right)$ contains a subword of the form $uXuXu$ with ${lim}_{n\to \infty }\left|{\phi }^n\left(u\right)\right|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words.

We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability ranks are inspired by the nonlinear indices à la Bourgain. In particular, we resolve a problem raised by F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht in Coarse and Lipschitz universality, Fund. Math. 254 (2021), no. 2, 181–214, regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result there.

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series ${\sum }_{n=1}^{\infty }\frac{a\left(n\right)}{n^s}$. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

We review some geometric criteria and prove a refined version, that yield existence of capillary surfaces in tubes $\Omega \times R$ in a gravity free environment, in the case of physical interest, that is, for bounded, open, and simply connected $\Omega \subset R^2$. These criteria rely on suitable weak one-sided bounds on the curvature of the boundary of the cross-section $\Omega$.

This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.

We answer a question of Samir Siksek, asked at the open problems session of the conference “Rational Points 2022”, which, in a broader sense, can be viewed as a reverse engineering of Diophantine equations. For any finite set $S$ of perfect integer powers, using Mihăilescu’s theorem, we construct a polynomial $f_S\in Z\left[x\right]$ such that the set $f_S\left(Z\right)$ contains a perfect integer power if and only if it belongs to $S$. We first discuss the easier case where we restrict to all powers with the same exponent. In this case, the constructed polynomials are inspired by Runge’s method and Fermat’s Last Theorem. Therefore we can construct a polynomial–exponential Diophantine equation whose solutions are determined in advance.

In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space $P^n\left(R\right)$ is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in $P^n\left(R\right)$ by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the centers with respect to a quadric. It is shown that the three reflections theorem, the three inversions theorem, notably Pascal’s theorem and Miquel’s theorem in Laguerre geometry are special cases of this theorem.

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu et al. (2021). In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.