Minkowski's Convex Body Theorem (with Blichfeldt's Lemma)

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Problem VIII.1.3 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.1 "Lebesgue Measure"

(a)(b)(c)

(a)

Prove that if $E \subset \mathbb {R}^{m}, \lambda_{m}(E)>1$ , then there exist two (distinct) points $x^{\prime }, x^{\prime \prime } \in E$ whose corresponding coordinate differences are integers.

(b)

Let $V \subset \mathbb {R}^{m}$ be a convex set, centrally symmetric about the origin, with $\lambda_{m}(V)>2^{m}$ . Prove that $V$ contains a point $a \neq 0$ with integer coordinates.

(c)

Let $V \subset \mathbb {R}^{m}$ be a convex set, centrally symmetric about the origin, with $\lambda_{m}(V)>N 2^{m}$ , $N$ being a positive integer. Prove that $V$ contains at least $2 N$ nonzero points with integer coordinates.

Problem VIII.6.26 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.6 "Asymptotics of High-Multiplicity Integrals"

Let $V_{n}$ be a convex set, centrally symmetric about zero, contained in the cube $(-1,1)^{n}$ . Prove that if $\lambda_{n}\left(V_{n}\right)>\left(\frac{3}{2}\right)^{n}$ , then for sufficiently large $n$ the set $2 V_{n}$ contains a point $a=\left(a_{1}, \ldots , a_{n}\right)$ with integer coordinates such that $\sum_{1 \leqslant k \leqslant n}\left|a_{k}\right|>\frac{n}{10}$ .

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.