Best Rational Approximations of an Irrational Number
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Preprint July 17 2018 Best rational approximations of an irrational number Jean-Louis Sikorav High Council for Economy Ministry for the Economy and Finance 120 rue de Bercy 75772 Paris cedex 12, France 2 Dedicated to the memory of Professor Denis Gerll 1913-2009 Abstract. Given an irrational number 훼 and an integer 푞 ≠ 0, there is a unique 1 integer 푝, equal to the nearest integer to 푞훼, such that |푞훼 − 푝| < . This result is 2 used to derive theorems and algorithms for the best approximations of an irrational by rational numbers, which are improved with the pigeonhole principle and used to offer an informed presentation of the theory of continued fractions. Introduction. The approximation of a real number by a rational one is an ancient problem encountered in various branches of knowledge, illustrated in astronomy by the theory of calendars and in engineering by the design of cogwheel astronomical clocks. The problem also arose in arithmetic with the discovery of irrational numbers, such as the square root of 푛 when 푛 is a positive integer that is not an exact square. This finding has given birth to the field of rational - also called Diophantine - approximations of such numbers. We know today that the set of rational numbers is - by construction - dense in the set of real numbers, so that for any given irrational number 훼, and for an arbitrary small number 휀 > 0, 푝 there exist infinitely many fractions , where 푝 and 푞 are integers with 푞 ≠ 0, that approximate 훼 푞 푝 with the degree of accuracy 휀: |훼 − | < 휀. Among these fractions, best rational approximations 푞 of an irrational number, abbreviated here as BRAIN, are the solutions of specific problems of extrema: 3 Given an irrational number 훼 and a positive integer 푛, find the integers (푥, 푦), with 0 < 푦 < 푛, minimizing 푥 either |훼 − | or |푦훼 − 푥|. 푦 푥 The corresponding fractions are the best rational approximations of 훼 having a denominator 푦 smaller in absolute value than a given quantity, known - using the terminology of Khinchin - as 푥 best approximations of the first kind (for the minimization of |훼 − |) and of the second kind 푦 (for the minimization of |푦훼 − 푥|). The first complete solution of these two problems is associated with the works of Wallis, Huygens, Euler and Lagrange and a few others. It relies on the apparatus of continued fractions, itself an offspring of Euclid’s algorithm, and provides existence theorems and algorithms for the computation of the approximations. The present investigation contains: 1) A solution of the problem of best rational approximations using a straightforward approach based on the concept of nearest integer, also yielding existence theorems and algorithms; 2) An improvement of the knowledge gained, through a confrontation with Dirichlet’s approximation theorem, leading to other theorems and algorithms; 3) An introduction to the field of continued fractions exploiting the results obtained in 1) and 2). 1. Nearest integer of an irrational number - Let 훼 be a positive irrational number. The set of natural integers greater than 훼 is not empty: according to the well-ordering principle, here equivalent to a principle of descent, it possesses a least element. This defines the least integer greater than 훼, denoted ⌈훼⌉. The number denoted ⌊훼⌋ = ⌈훼⌉ − 1 is in turn the greatest integer less than 훼. Both symbols have been conceived by Iverson. The two numbers ⌊훼⌋ and ⌈훼⌉ are the only integers 푛 such that |훼 − 푛| < 1. 4 If 훼 is a negative irrational number, – 훼 is positive. We can write: ⌊−훼⌋ <– 훼 < ⌈−훼⌉, hence: −⌈−훼⌉ < 훼 < −⌊−훼⌋, showing in that case too the existence of a greatest integer less than 훼 and a least integer greater than 훼, with ⌊훼⌋ = −⌈−훼⌉ and ⌈훼⌉ = −⌊−훼⌋. This establishes the existence of such integers for all irrational numbers. - We define the fractional part {훼} = 훼 − ⌊훼⌋, with 0 < {훼} < 1. This number measures the distance between 훼 and ⌊훼⌋ and the number 1 − {훼} the distance between 훼 and ⌈훼⌉. Of these 1 two numbers, only one is strictly smaller than , as their sum is equal to 1 and the equality 2 1 − {훼} = {훼} is ruled out by the irrationality of 훼. Accordingly, of the two numbers ⌊훼⌋ and ⌈훼⌉, only one, called the nearest integer to 훼 and denoted here by the more symmetrical symbol [훼] (Gauss square bracket) is such that: 1 0 < |훼 − [훼] | < . 2 1 1 The nearest integer verifies: [훼] = ⌊훼 + ⌋ = ⌈훼 − ⌉. 2 2 - The distance between 훼 and the nearest integer is written ‖훼‖, with: ‖훼‖ = |훼 − [훼] | = min({훼}, 1 − {훼}). 2. Nearest rational of an irrational number, irrationality criteria and representations of irrational numbers Let 푞 be a strictly positive integer. The preceding results can be generalized through the replacement of 훼 by the irrational number 푞훼: - There are exactly two integers 푝, equal to ⌊푞훼⌋ and ⌈푞훼⌉, such that: |푞훼 − 푝| < 1. For the first integer |푞훼 − ⌊푞훼⌋ | = {푞훼} and for the second |푞훼 − ⌈푞훼⌉| = 1 − {푞훼}. - There is a unique integer 푟 = [푞훼] such that: 1 0 < |푞훼 − 푟| = ‖푞훼‖ < . 2 5 This leads to basic irrationality criteria: Let 훼 be a real number. The following statements are equivalent: (i) 훼 is irrational; (ii) For any strictly positive integer 푞, there are exactly two integers 푝, such that: |푞훼 − 푝| < 1; (iii) For any strictly positive integer 푞, there is a unique integer 푟 such that: 1 0 < |푞훼 − 푟| = ‖푞훼‖ < . 2 Dividing by 푞 the members of the inequalities above further gives: - There are exactly two integers 푝, such that: 푝 1 |훼 − | < . 푞 푞 - There is a unique integer 푟 such that: 푟 ‖푞훼‖ 1 0 < |훼 − | = < . 푞 푞 2푞 These results lead to further irrationality criteria: Let 훼 be a real number. The following statements are equivalent: (i) 훼 is irrational; (ii) For any strictly positive integer 푞, there are exactly two integers 푝 such that: 푝 1 |훼 − | < ; 푞 푞 (iii) For any strictly positive integer 푞, there is a unique integer 푟 such that: 푟 1 0 < |훼 − | < . 푞 2푞 In this last inequality, one cannot lower the denominator 2푞 (say by replacing the factor 2 by (2 − 휀) or 푞 by 푞1−휖, with 0 < 휀 < 1) without losing the uniqueness of 푟. 6 [푞훼] 1 The sequence ( ) is such that the distance between its 푞th term and 훼 is smaller ; thus it 푞 푞≥1 푞 [푞훼] converges towards 훼 when 푞 goes to infinity: lim = 훼. The same conclusions apply to the 푞→∞ 푞 ⌊푞훼⌋ ⌈푞훼⌉ sequences of underestimates ( ) and of overestimates ( ) of 훼 . These three 푞 푞≥1 푞 푞≥1 sequences are two by two distinct and each provides a unique representation of 훼. [푞훼] The rational number is the nearest rational fraction to 훼 for an arbitrary numerator and the 푞 fixed denominator 푞 . Stated otherwise, it is the best rational approximation of 훼 for this denominator. To acknowledge the ease with which this best approximation is derived, we shall call it 푇푅퐴퐼푁 (훼, 푞) (for Trivial Approximation of an Irrational Number). It will now be used to obtain best approximation theorems and algorithms. 3. TRAIN to BRAIN 3.1 BRAIN Theorems First BRAIN theorem. Let 훼 be an irrational number. There is a unique, infinite sequence of 푝푘 rationals ( )푘≥1, where 푝푘 and 푞푘 are relatively prime integers with 1 ≤ 푞푘 and 푝푘 = [푞푘훼], 푞푘 푝 ‖푞훼‖ 푝 푝 푝 such that |훼 − 푘 | = min and |훼 − 푘+1 | < |훼 − 푘 |. The fraction 푘 is the 푘푡ℎ best 푞푘 1≤푞≤푞푘 푞 푞푘+1 푞푘 푞푘 rational approximation of the first kind of 훼 and denoted 퐵푅퐴퐼푁 퐼(훼, 푘). The sequence (퐵푅퐴퐼푁 퐼(훼, 푘))푘≥1 converges to 훼 when 푘 → ∞ and provides a unique representation of 훼. Proof. Let 푁 be a strictly positive integer. For each integer 푞 with 1 ≤ 푞 ≤ 푁, the smallest value of 푝 ‖푞훼‖ 푝 |훼 − | where 푝 is an integer, is equal to and is obtained for 푝 = [푞훼]: the fraction is 푞 푞 푞 푝 simply 푇푅퐴퐼푁 (훼, 푞). This implies that the set of numbers of the form |훼 − |, where 푝 and 푞 푞 are integers with 1 ≤ 푞 ≤ 푁 possesses a smallest element equal to the smallest of the 푁 numbers 7 ‖푞훼‖ ‖푞훼‖ with 1 ≤ 푞 ≤ 푁. This number, min , is obtained for one (or several) integer(s) 푞. Let 푞 1≤푞≤푁 푞 us call 푞0 the smallest denominator. If the two numbers [푞0훼] and 푞0 are not relatively prime, then we can write [푞0훼] = 푢푠 and 푞0 = 푢푡, where 푠, 푡 and 푢 are integers with 푢 ≥ 2, showing 푠 ‖푞훼‖ that |훼 − | is also equal to min , with 푡 < 푞0, in contradiction with the definition of 푞0. 푡 1≤푞≤푁 푞 1 [푞0훼] The two numbers [푞0훼] and 푞0 are thus relatively prime. The fraction is therefore 푞0 irreducible and is the best rational approximation of the first kind with a denominator ≤ 푁. For 푁 = 1, the first term of the sequence 퐵푅퐴퐼푁 퐼(훼, 푘), 퐵푅퐴퐼푁 퐼(훼, 1) is given by 푞1 = 1 and [푞훼] 푝 = [훼]. Since the sequence ( ) converges towards 훼 when 푞 → ∞, there is an integer 1 푞 푞≥1 ′ ′ [푛 훼] [푞0훼] 푛 > 푁 such that |훼 − ′ | < |훼 − |. This shows that the sequence (퐵푅퐴퐼푁 퐼(훼, 푘))푘≥1 푛 푞0 is unbounded and converges to 훼 when 푘 → ∞. Lastly, one can see that by construction, 푘′ > 푘 푝푘′ 푝푘 ‖푞훼‖ implies 푞푘′ > 푞푘 and |훼 − | < |훼 − |: the sequence min strictly decreases to zero 푞푘′ 푞푘 1≤푞≤푞푘 푞 when 푞푘 → ∞.