Math 620  Problem Set #10  Due Mon., Nov. 26, 2001

1. If a, b ∈ ℤ are relatively prime and b > 0, let I(a/b) be the open interval given by \frac{a}{b} − \frac{1}{\sqrt{5}b^{2}} < x < \frac{a}{b} + \frac{1}{\sqrt{5}b^{2}} . Let Ī(a/b) be the correponding closed interval.

a) Let a/b, c/d be consecutive Farey fractions in some F_{n}. Show that

I(a/b) ∩ I(c/d) ≠ ∅ ⇔ Ī(a/b) ∩ Ī(c/d) ≠ ∅ ⇔ b/d + d/b ≥ \sqrt{5}.

[Hint: What is c/d − a/b?]

b) Let e/f be the first Farey fraction to occur between a/b, c/d in a later Farey sequence F_{m}. Show that the intersections I(a/b) ∩ I(c/d) and I(a/b) ∩ I(e/f) cannot both be empty. [Hint: Use part (a) and PS 4 #2c, and then show that for any real number t > 0, either f(t) ≥ \sqrt{5} or f(1 + t) ≥ \sqrt{5}, where f(x) = x + \frac{1}{x}.] Deduce by symmetry that I(c/d) must meet either I(a/b) or I(e/f).

c) Deduce that I(a/b) ∪ I(e/f) ∪ I(c/d) contains the open interval (a/b, c/d).

d) Deduce that if α is a real number in the open interval (a/b, c/d), then one of the fractions p/q = a/b, c/d, e/f satisfies |α − p/q| < 1/\sqrt{5}q^{2}.

e) Conclude from (d) that for any irrational number α, there are infinitely may fractions p/q such that |α − p/q| < 1/\sqrt{5}q^{2}.

2. Let S be the set of irrational numbers α whose continued fraction expansion [a_{0}, a_{1}, …] has the property that a_{n} ∈ {1, 2} for all n ≫ 0. Let S′ be the complement of S in the set of all irrational numbers. Let M(α) be the Markov constant of an irrational number α.

a) Evaluate a = sup{M(α) | α ∈ S}. Is this supremum achieved — i.e. is there an α ∈ S with M(α) = a? If so, find such an α. If not, find a sequence α_{n} in S such that M(α_{n}) → a.

b) Evaluate b = inf{M(β) | β ∈ S′}. Is this infimum achieved — i.e. is there a β ∈ S′ with M(β) = b? If so, find such a β. If not, find a sequence β_{n} in S′ such that M(β_{n}) → b.

3. If Λ is a set of points in the plane, call Λ a lattice if there are two linearly independent vectors v, w ∈ ℝ^{2} with Λ = {mv + nw | m, n ∈ ℤ}. The fundamental parallelogram of Λ is the parallelogram with vertices 0, v, w, v + w.

a) Generalize Problem Set 9 #4d to the case in which the lattice ℤ^{2} is replaced by an arbitrary lattice Λ. Express your answer in terms of the area Δ of the fundamental parallelogram of Λ.

b) Generalize part (a) from ℝ^{2} to ℝ^{n}, where a lattice in ℝ^{n} is generated by n linearly independent vectors.

4. a) Suppose p is a prime and a^{2} + 1 ≡ 0 (mod p). Let Λ be the set of points in the plane with coordinates (u_{1}, u_{2}) such that u_{1}, u_{2} ∈ ℤ and u_{1} ≡ au_{2} (mod p). Draw Λ for p = 5, 13. Can one do this for p = 7?

b) Find two vectors that generate Λ, and find the area of the fundamental parallelogram they span.

c) Consider the convex symmetric region x^{2} + y^{2} < 2p. Find all the points of Λ that are in the region. Do this for p = 5, 13. Do such points exist? What is their distance from the origin. Conjecture? Proof? [Hint: Problem 3.]

d) Using (c), give a new proof that every prime ≡ 1 (mod 4) is a sum of two squares.