### Working paper

## Limit complexities revisited [once more]

Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad ("non-stochastic") data appear "in real life"? Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory.

We present a new structural lemma for deterministic con- text free languages. From the first sight, it looks like a pumping lemma, because it is also based on iteration properties, but it has significant distinctions that makes it much easier to apply. The structural lemma is a combinatorial analogue of KC-DCF-Lemma (based on Kolmogorov complexity), presented by Li and Vit ́anyi in 1995 and corrected by Glier in 2003. The structural lemma allows not only to prove that a language is not a DCFL, but discloses the structure of DCFLs Myhill-Nerode classes.

The paper [Harry Buhrman, Michal Kouck ́, Nikolay Vereshcha- y gin. Randomized Individual Communication Complexity. IEEE Con- ference on Computational Complexity 2008: 321-331] considered com- munication complexity of the following problem. Alice has a bi- nary string x and Bob a binary string y, both of length n, and they want to compute or approximate Kolmogorov complexity C(x|y) of x conditional to y. It is easy to show that deterministic communica- tion complexity of approximating C(x|y) with precision α is at least n − 2α − O(1). The above referenced paper asks what is random- ized communication complexity of this problem and shows that for r- round randomized protocols its communication complexity is at least Ω((n/α)1/r ). In this paper, for some positive ε, we show the lower bound 0.99n for (worst case) communication length of any random- ized protocol that with probability at least 0.01 approximates C(x|y) with precision εn for all input pairs.

Antistochastic strings are those strings that do not have any reasonable statistical explanation. We establish the follow property of such strings: every antistochastic string *x* is “holographic” in the sense that it can be restored by a short program from any of its part whose length equals the Kolmogorov complexity of *x*. Further we will show how it can be used for list decoding from erasing and prove that Symmetry of Information fails for total conditional complexity.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.