Problem 1We shall work entirely in a vector space V over the rational number. For r ∈ Q, we

define +r : V 2 −→ V by

+r (a, b) = a + rb.

We add the definition

+∞(a, b) = b.

We continue to use − as +−1.

We use the notation SD(r1, . . . , rm; α) to be the statement that if G ⊂ V 2 is finite with | +rj

(G)| ≤ N for all j = 1, . . . , m but with − one to one on G then |G| 1 +

√

2

2 there is a choice of a finite list of rationals

r1 , . . . , rm different from −1 with SD(r1 , . . . , rm ; α). Hint: Take one of the functions νt,r

from Problem 3 and apply whatever SD result you might know to an appropriately chosen

level set. Repeat as needed.