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V. Bernik

Publications and source records attributed to V. Bernik.

4 recordsLinked to original sources
arXiv2017

On distribution of points with conjugate algebraic integer coordinates close to planar 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 algebraic conjugate integer coordinates of degree $\leq n$ and of height $\leq Q$. Denote by $\tilde{M}^n_φ(Q,γ, J)$ the set of points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. In this paper we show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, which are independent of $Q$, such that $c_2\cdot Q^{n-γ}<# \tilde{M}^n_φ(Q,γ, J)< c_3\cdot Q^{n-γ}$.

math.NT↗
arXiv2016

On points with algebraically conjugate coordinates close to smooth curves

We show that for any sufficiently large integer $Q$ and a real $0\leqλ\leq\frac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,λ)=\{(x,y):|y-f(x)|<Q^{-λ}, x\in J=[a,b]\}$ contain at least $c(n, f, J)Q^{n+1-λ}$ points $\barγ=(α,β)$ with algebraically conjugate coordinates. We consider points $\barγ$ such that the minimal polynomial $P(x)$ of $α,β$ is of degree $°P\leq n,\ n\ge 2$, and height $H(P)\leq Q$. The proof is based on a metric theorem on the measure of the set of vectors $(x,y)$ lying in a rectangle $Π$ of dimensions $Q^{-s_1}\times Q^{-s_2}$ with $|P(x)|, |P(y)|$ bounded from above and $|P'(x)|,|P'(y)|$ bounded from below, where $P(x)$ is a polynomial of degree $°P\leq n$ and height $H(P)\leq Q$. This theorem is a generalization of a result obtained by V. Bernik, F. Götze and O. Kukso for $s_1=s_2=\frac12$ and $λ= \frac12$.

math.NT↗