4400/6400 PROBLEM SET 7
Recommendation:
4400 students should do at least 4 problems; 6400 students
should do at least 6.
7.1) Write out a careful proof of Lemma 6 in the Minkowski’s Theorem handout.
Especially, say a bit about the changeofvolume properties of the determinant map.
Citing a reference or two might be appropriate.
7.2)** Prove Theorem 11 in the Minkowski’s theorem handout.
(Possible strat
egy: talk to Professor Joseph H.G. Fu.)
7.3) Let Ω
⊂
R
N
be a convex body with volume exactly 2
N
.
(We have seen
that Ω need not have a nonzero lattice point.) Show that if Ω is moreover
closed
,
then it does have a nonzero lattice point.
1
Suggestion:
Use the fact that for all
>
0, the dilate (1 +
)Ω must contain a
nonzero lattice point.
7.4)* Prove Theorem 12 in the Minkowksi’s Theorem handout.
7.5) Prove Lemma 14 (Euler’s identity) in the Minkowksi’s Theorem handout.
7.6) Prove Lemma 16 in the Minkowksi’s Theorem handout.
7.7) The
three squares theorem
of LegendreGauss says that a positive inte
ger
n
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 Spring '11
 Staff
 Number Theory, Addition, Natural number, Prime number, nonzero lattice point

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