• Madhu Raka

Articles written in Proceedings – Mathematical Sciences

• Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Гr,n—r denote the infimum of all number Г &gt; 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c1; c2,…, cn, there exist integersx1,x2,…,xn satisfying 0 &lt; Q(x1+c1,x2 + c2,…,xn + cn) ≤(Г¦Z &gt; ¦)1/n. All the values of Гr,n—r are known except for г1,4. Earlier it was shown that 8 ≤Г1,4 ≤16. Here we improve the upper bound to get Г1,4 &lt; 12.

• On conjectures of Minkowski and Woods for $n = 9$

Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant 1 such that there is a sphere $\mid X \mid \lt R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly independent points of $\wedge$ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in $\mathbb{R}^n$ of radius $\sqrt{n/4}$ contains a point of $\wedge$. This is known to be true for $n\leq 8$. Here we prove a more general conjecture of Woods for $n = 9$ from which this conjecture follows in $\mathbb{R}^9$. Together with a result of McMullen (J. Amer. Math. Soc. 18 (2005) 711–734), the long standing conjecture of Minkowski follows for $n = 9$.

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