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.
