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

Wojciech Banaszczyk; Artur Lipnicki

On the lattice of polynomials with integer coefficients: the covering radius in

L_{p} (0, 1)

Wojciech Banaszczyk ; Artur Lipnicki

Annales Polonici Mathematici, Tome 113 (2015), p. 123-144 / Harvested from The Polish Digital Mathematics Library

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Résumé

The paper deals with the approximation by polynomials with integer coefficients in $L_{p} (0, 1)$ , 1 ≤ p ≤ ∞. Let $P_{n, r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^{r} {(1 - x)}^{r}$ , r ≥ 0, and let $P_{n, r}^{ℤ} \subset P_{n, r}$ be the set of polynomials with integer coefficients. Let $μ (P_{n, r}^{ℤ}; L_{p})$ be the maximal distance of elements of $P_{n, r}$ from $P_{n, r}^{ℤ}$ in $L_{p} (0, 1)$ . We give rather precise quantitative estimates of $μ (P_{n, r}^{ℤ}; L ₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $μ (P_{n, r}^{ℤ}; L_{p})$ for p ≠ 2. It follows that $μ (P_{n, r}^{ℤ}; L_{p}) ≍ n^{- 2 r - 2 / p}$ as n → ∞. The results partially improve those of Trigub [Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962)].

Publié le : 2015-01-01

Zbl 1336.41003

EUDML-ID : urn:eudml:doc:280296

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     author = {Wojciech Banaszczyk and Artur Lipnicki},
     title = {On the lattice of polynomials with integer coefficients: the covering radius in $L\_p(0,1)$
            },
     journal = {Annales Polonici Mathematici},
     volume = {113},
     year = {2015},
     pages = {123-144},
     zbl = {1336.41003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap115-2-2}
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Wojciech Banaszczyk; Artur Lipnicki. On the lattice of polynomials with integer coefficients: the covering radius in $L_p(0,1)$
            . Annales Polonici Mathematici, Tome 113 (2015) pp. 123-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap115-2-2/