On van der Corput property of shifted primes

We prove that the upper bound for the van der Corput property of the set of shifted primes is O((log n)^{-1+o(1)}), giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes p-1. We construct normed non-negative valued cosine polynomials with the spectrum in the set p-1, p<=n, and a small free coefficient a_0=O((log n)^{-1+o(1)}). This implies the same bound for the Poincar\'e property of the set p-1, and also bounds for several properties related to uniform distribution of related sets

A new measure of instability and topological entropy of area-preserving twist diffeomorphisms

We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp >0 lower bound on the topological entropy in a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming only no topological obstruction to diffusion, i.e. no homotopically non-trivial invariant circle consisting of orbits with the rotation number 0. The proof is based on a new method of precise construction of positive entropy invariant measures, applicable to more general Lagrangian systems, also in higher degrees of freedom

On van der Corput property of squares

We prove that the upper bound for the van der Corput property of the set of perfect squares is O((log n)^{-1/3}), giving an answer to a problem considered by Ruzsa and Montgomery. We do it by constructing non-negative valued, normed trigonometric polynomials with spectrum in the set of perfect squares not exceeding n, and a small free coefficient a_0=O((log n)^{-1/3})

Anthropocene, Capitalocene, Machinocene: Illusions of instrumental reason

In their seminal work, Dialectics of Enlightenment, Horkheimer and Adorno interpreted capitalism as the irrational monetization of nature. In the present work, I analyze three 21st century concepts, Anthropocene, Capitalocene and Machinocene, in light of Horkheimer and Adorno’s arguments and recent arguments from the philosophy of biology. The analysis reveals a remarkable prescience of the term “instrumental reason”, which is present in each of the three concepts in a profound and cryptic way. In my interpretation, the term describes the propensity of science based on the notion of physicalism to interpret nature as the machine analyzable and programmable by the human reason. As a result, the Anthropocene concept is built around the mechanicist model, which may be presented as the metaphor of the car without brakes. In a similar fashion, the Machinocene concept predicts the emergence of the mechanical mind, which will dominate nature in the near future. Finally, the Capitalocene concept turns a perfectly rational ambition to expand knowledge into an irrational obsession with over-knowledge, by employing the institutionalized science as the engine of capitalism without brakes. The common denominator of all three concepts is the irrational propensity to legitimize self-destruction. Potential avenues for countering the effects of “instrumental reason” are suggested

Positive exponential sums and odd polynomials

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient a_{0}=O((\log n)^{-1/k}). This gives an alternative proof for the maximal possible cardinality of a set A, so that A-A does not contain an element of f(x). We also discuss other interpretations and an ergodic characterization of that bound