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Problems for Chapter III
Problems for Chapter III
2
2
2
1. Show that the number of lattice points in x + y ≤ n , x, y ≥ 0,
π 2
is ∼ 4 n . By the Riemann integral method show that it is, in fact
π 2
n + O(n).
4
√
2. If x is bounded by its own square root (i.e., by x + a), then we
find that it has a pure bound. What if x, instead, is bounded by
x 2/3 + ax 1/3 + b? Does this insure a bound on x?
3. Suppose that a convex closed curve has its curvature bounded by
√
δ. Show that it must come within 2 δ of some lattice point.
4. Produce a convex closed curve with curvature bounded by δ which
√
doesn’t come within δ of any lattice point.
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