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Consider a real matrix consisting of rows for . The problem of making the system of linear forms for integers , small naturally induces an ordinary and a uniform exponent of approximation, denoted by and respectively. For , a sharp lower bound for the ratio was recently established by Marnat and Moshchevitin. We give a short, new proof of this result upon a hypothesis on the best approximation integer vectors associated to . Our bound applies to general , but is probably not optimal in this case. Thereby we also complement a similar conditional result of Moshchevitin, who imposed a different assumption on the best approximations. Our hypothesis is satisfied in particular for , and thereby unconditionally confirms a previous observation of Jarník. We formulate our results in a very general context of approximation of subspaces of Euclidean spaces by lattices. We further establish criteria upon which a given number of consecutive best approximation vectors are linearly independent. Our method is based on Siegel’s lemma.
A spiral in is defined as a set of the form , where is a spherical sequence. Such point sets have been extensively studied, in particular in the planar case , as they then serve as natural models describing phyllotactic structures (i.e., structures representing configurations of leaves on a plant stem).
Recent progress in this theory provides a fine analysis of the distribution of spirals (e.g., their covering and packing radii). Here, various concepts of visibility from discrete geometry are employed to characterise density properties of such point sets. More precisely, necessary and sufficient conditions are established for a spiral to be an orchard (a “homogeneous” density property defined by Pólya), a uniform orchard (a concept introduced in this work), a set with no visible point (implying that the point set is dense enough in a suitable sense) and a dense forest (a quantitative and uniform refinement of the previous concept).
We simplify and improve the constant that appears in effective irrationality measures,
obtained from the hypergeometric method for near . The dependence of on in our result is best possible (as is the dependence on in many cases). For some applications, the dependence of this constant on becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.
We prove a series of new results for the irrationality measure of some values of . Some time ago we applied symmetric complex integrals to approximate values of the form , where is a natural number. This method gave new estimates for the numbers and only. To deal with some other values of we modify the main construction. In the present paper, we consider a new integral which is based on an idea of Q. Wu and does not have a property of symmetry of the integrand. Integral construction of such a type allows us to improve estimates for the irrationality measure of some values of for some natural , .
This note pushes further the discussion by Beresnevich, Guan, Marnat, Ramirez, and Velani (Adv. Math.401 (2022), art. id. 108316) about relations between Dirichlet improvable, badly approximable and singular points by considering Diophantine sets extending the notion of bad approximability.