Number Theory: An Introduction to Mathematics

190 IV Continued Fractions and Their Uses

thenan=1 for all largen. The constant

√

8 is again best possible, since a similar
argument to that just given shows that ifσ := 1 +

√

2 =[2, 2 ,...] then, for any
c>

√

8, there exist at most finitely many rational numbersp/qsuch that

|σ−p/q|< 1 /cq^2.

It follows from Proposition 6 withr=3thatif

|ξ−pn/qn|≥ 1 /

√

13 q^2 n for all largen,

thenan∈{ 1 , 2 }for all largen.
For any irrationalξ, with continued fraction expansion [a 0 ,a 1 ,...] and conver-
gentspn/qn, put

M(ξ)= lim

n→∞

qn−^1 |qnξ−pn|−^1.

It follows from Proposition 2 thatM(ξ)=M(η)ifξandηare equivalent. The results
just established show thatM(ξ)≥

√

5foreveryξ.IfM(ξ) <

√

8, thenan=1forall
largen; henceξis equivalent toτandM(ξ)=M(τ)=

√

5. IfM(ξ) <

√

13, then
an∈{ 1 , 2 }for all largen.
An irrational numberξis said to bebadly approximableifM(ξ) < ∞.The
inequalities

an+ 2 /qnqn+ 2 <|ξ−pn/qn|< 1 /qnqn+ 1

imply

an+ 1 ≤qn+ 1 /qn<qn−^1 |qnξ−pn|−^1

and

q−n^1 |qnξ−pn|−^1 <qn+ 2 /an+ 2 qn≤qn+ 1 /qn+ 1 ≤an+ 1 + 2.

Henceξis badly approximable if and only if its partial quotients anare bounded.
It is obvious thatξis badly approximable if there exists a constantc>0 such that

|ξ−p/q|>c/q^2

for every rational numberp/q.Conversely,ifξ is badly approximable, then there
exists such a constantc>0. This is clear whenpandqare coprime integers, since if
p/qisnotaconvergentofξthen, by Proposition 4,

|ξ−p/q|≥ 1 / 2 q^2.

On the other hand, ifp=λp′,q=λq′,wherep′,q′are coprime, then

|ξ−p/q|=|ξ−p′/q′|≥c/q′^2 =λ^2 c/q^2 ≥c/q^2.

Some of the applications of badly approximable numbers stem from the following
characterization: a real numberθis badly approximable if and only if there exists a
constantc′>0suchthat

|e^2 πiqθ− 1 |≥c′/q for allq∈N.