Diophantine Approximations
By Ivan Niven
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The author refrains from the use of continuous fractions and includes basic results in the complex case, a feature often neglected in favor of the real number discussion. Each chapter concludes with a bibliographic account of closely related work; these sections also contain the sources from which the proofs are drawn.
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Diophantine Approximations - Ivan Niven
end.
CHAPTER 1
The Approximation of Irrationals by Rationals
1.1. The pigeon-hole principle
Given a real number θ, how closely can it be approximated by rational numbers? To make the question more precise, for any given positive ε is there a rational number a/b within ε of θ, so that the inequality
is satisfied? The answer is yes because the rational numbers are dense on the real line. In fact, this establishes that for any real number θ and any positive ε there are infinitely many rational numbers a/b satisfying the above inequality.
Another way of approaching this problem is to consider all rational numbers with a fixed denominator b, where b is a positive integer. The real number θ can be located between two such rational numbers, say
and so we have |θ − c/b| < 1/b. In fact, we can write
(1)
by choosing a = c or a = c + 1, whichever is appropriate. The inequality (1) would be strict, that is to say, equality would be excluded if θ were not only real but irrational. We shall confine our attention to irrational numbers θ because most of the questions about approximating rationals by rationals reduce to simple problems in linear Diophantine equations.
Now by use of the pigeon-hole principle (sometimes called the box principle) we can improve inequality (1) as in the following theorem. The pigeon-hole principle states that if n + 1 pigeons are in n holes, at least one hole will contain at least two pigeons.
THEOREM 1.1. Given any irrational number θ and any positive integer m, there is a positive integer b ≦ m such that
The symbol a here denotes the integer nearest to bθ, so that the equality ∥bθ∥ = |bθ − a| holds by the definition of the symbolism.
Proof: Consider the m + 2 real numbers
(2)
lying in the closed unit interval. Divide the unit interval into m + 1 subintervals of equal length
(3)
Since θ is irrational, each of the numbers (2) except 0 and 1 lies in the interior of exactly one of the intervals (3). Hence two of the numbers (2) lie in one of the intervals (3); thus there are integers k1, k2, h1, and h2 such that
We may presume that m ≧ k2 > k1 ≧ 0. Defining b = k2 − k1, a = h2 − h1, we have established the theorem.
Since (m + 1)−1 < b−1, Theorem 1.1 implies that ∥bθ∥ < b−1. Furthermore, this inequality is satisfied by infinitely many positive integers b for the following reason. Suppose there were only a finite number of such integers, say b1, b2, ⋯, br with
Then choose the integer m so large that
holds for every j = 1, 2, ⋯, r. Then apply Theorem 1.1 with this value of m, and note that this process yields an integer b such that
Hence b is different from each of b1, b2, . . . , br. Also ∥bθ∥ < b−1, so there can be no end to the integers satisfying this inequality. The following corollary states what we have just proved.
COROLLARY 1.2. Given any irrational number θ, there are infinitely many rational numbers a/b, where a and b > 0 are integers, such that
(4)
Note that this result is a considerable improvement over inequality (1). It is natural to ask whether Corollary 1.2 can also be improved, for instance, by the replacement of 1/b² by 1/b³. It cannot; in fact, Corollary 1.2 becomes false if 1/b² is replaced by 1/b²+ε for any positive ε. Nevertheless, although the exponent cannot be improved, this corollary can be strengthened by a constant factor in (4). Specifically 1/b, and no larger constant can be used than √5. This result, due to Hurwitz, is proved in the next section.
1.2. The theorem of Hurwitz
We first prove a preliminary result about Farey sequences. For any positive integer n, the Farey sequence Fn is the sequence, ordered in size, of all rational fractions a/b in lowest terms with 0 < b ≦ n. For example,
Of the many known properties of Farey sequences, only two are needed for our purposes, as follows.
THEOREM 1.3. If a/b and c/d are two consecutive terms in Fn, then, presuming a/b to be the smaller, bc − ad = 1. Furthermore, if θ is any given irrational number, and if r is any positive integer, then for all n sufficiently large the two fractions a/b and c/d adjacent to θ in Fn have denominators larger than r, that is, b > r and d > r.
Proof: The proof of the first part is by induction on n. If n = 1, then b = 1, d = 1, and c = a + 1, so that
bc − ad = a + 1 − a = 1.
Next we suppose that the result holds for Fn, and prove it for Fn+1. Let a/b and c/d be adjacent fractions in Fn. First we note that b + d ≧ n + 1, since otherwise the fraction (a + c)/(b + d), reduced if necessary, would belong to Fn. But this is not possible