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Vasili Bernik

Publications and source records attributed to Vasili Bernik.

5 recordsLinked to original sources
arXiv2017

Systems of small linear forms and Diophantine approximation on manifolds

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprindžuk from the 1970s on approximations to several real numbers by conjugate algebraic numbers. Our main result is a Khintchine type theorem, which convergence case is established without usual monotonicity constrains and the divergence case is proved for Hausdorff measures. The result encompasses several previous findings and, within the setup considered, gives the best possible improvement of a recent theorem of Aka, Breuillard, Rosenzweig and Saxcé on extremality.

math.NT↗
arXiv2017

On distribution of points with algebraically conjugate coordinates in neighborhood of smooth curves

Let $φ:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbolα=(α_1,α_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $α_1,α_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_φ(Q,γ, J)$ the set of such points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. We show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, where $c_i=c_i(n)$, $i=1,2$, which are independent of $Q$, such that $c_2 Q^{n+1-γ}<\# M^n_φ(Q,γ, J)< c_3 Q^{n+1-γ}$.

math.NT↗
arXiv2015

Integral polynomials with small discriminants and resultants

Let $n\in\mathbb{N}$ be fixed, $Q>1$ be a real parameter and $\mathcal{P}_n(Q)$ denote the set of polynomials over $\mathbb{Z}$ of degree $n$ and height at most $Q$. In this paper we investigate the following counting problems regarding polynomials with small discriminant $D(P)$ and pairs of polynomials with small resultant $R(P_1,P_2)$: (i) given $0\le v\le n-1$ and a sufficiently large $Q$, estimate the number of polynomials $P\in\mathcal{P}_n(Q)$ such that $$0<|D(P)|\le Q^{2n-2-2v};$$ (ii) given $0\le w\le n$ and a sufficiently large $Q$, estimate the number of pairs of polynomials $P_1,P_2\in\mathcal{P}_n(Q)$ such that $$0<|R(P_1,P_2)|\le Q^{2n-2w}.$$ Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R.C.Vaughan and S.Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

math.NT↗
arXiv2010

The distribution of close conjugate algebraic numbers

We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds of Bugeaud and Mignotte. So far the results a la Bugeaud and Mignotte relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. The applications of our main theorem considered in this paper include generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

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