On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$

Wojciech Banaszczyk¹ ; Artur Lipnicki¹

¹ Faculty of Mathematics and Computer Science University of Łódź 90-238 Łódź, Poland

Annales Polonici Mathematici, Tome 115 (2015) no. 2, pp. 123-144

Résumé

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, $1\le p\le \infty $. Let $\boldsymbol {P}_{n,r}$ be the space of polynomials of degree $\le n$ which are divisible by the polynomial $x^r(1-x)^r$, $r\ge 0$, and let $\boldsymbol {P}_{n,r}^\mathbb {Z}\subset \boldsymbol {P}_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)$ be the maximal distance of elements of $\boldsymbol {P}_{n,r}$ from $\boldsymbol {P}_{n,r}^\mathbb {Z}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_2)$ for $n\gtrsim 6r$. Then we obtain similar, somewhat less precise, estimates of $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)$ for $p\not =2$. It follows that $\mu (\boldsymbol {P}_{n,r}^\mathbb {Z};L_p)\asymp n^{-2r-2/p}$ as $n\to \infty $. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].

Détail

DOI : 10.4064/ap115-2-2

Keywords: paper deals approximation polynomials integer coefficients infty boldsymbol space polynomials degree which divisible polynomial x boldsymbol mathbb subset boldsymbol set polynomials integer coefficients boldsymbol mathbb maximal distance elements boldsymbol boldsymbol mathbb rather precise quantitative estimates boldsymbol mathbb gtrsim obtain similar somewhat precise estimates boldsymbol mathbb follows boldsymbol mathbb asymp r infty results partially improve those trigub izv akad nauk sssr ser mat