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Generalization and Proof of the Littlewood Conjecture

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Generalization and Proof of the Littlewood Conjecture American Journal of Mathematics and Statistics 2017, 7(3): 113-124 DOI: 10.5923/j.ajms.20170703.04 Generalization and Proof of the Littlewood Conjecture Kaveh Mozafari ExcellenSation Inc., Toronto, Ontario, Canada Abstract The aim of this paper is to provide proof of the Littlewood conjecture (theorem). The Littlewood conjecture proposed by John Edensor Littlewood close to 90 years ago. There are numerous applications of the conjecture to this day; however, a complete proof over real number is yet to be provided. The conjecture proposes that the limit inferior (roughly speaking the lower bound of limit) of multiplication between an infinitely large number, n, and the distances between two real numbers each multiplied by n, from their closest integer approaches zero. This paper provides a proof of the Littlewood conjecture. The proof segregates the real numbers in two forms of rational and irrational numbers. All parts of the proof utilize the analytical approach of limit. Initially, a proof of rational numbers is provided. In consequent to such proof, the validity of the conjecture is revealed by the Archimedean's property of real number followed by squeeze theorems for all real numbers. After such claim, it would be evident that the Littlewood conjecture is a special form of a bigger limit inferior which approaches zero. Keywords Littlewood conjecture proof, Littlewood conjecture generalization, Diophantine approximation (set of all real numbers) means the distance of b from the 1. Introduction closest integer while αβ, ∈ . Moreover liminfnn→∞ x Littlewood Conjecture is an interesting conjecture in is defined as follow. mathematics which was proposed by John Edensor liminfxxnm := lim(inf ) Littlewood close to 90 years ago. The conjecture considered n→∞ n→∞ mn ≥ an open problem in mathematics since 2016. The presented In the present paper the Littlewood conjecture is proven paper proves the conjecture while it generalizes it to a to be correct by first proving the conjecture for rational theorem which states that limit inferior of multiplication of a number and then extending it to all real numbers. At the end finite number of real number's distance from the nearest the conjecture is extended to cover more terms as follow. integer is always zero. By limit inferior, we are referring to liminfnn ||αα12 |||| n ||...|| n αp ||= 0, p ∈ the lower bound of the limit upon its existence. For example n→∞ limit inferior of cosine function for real value would be -1, To get limit inferior, we need to realize where the lowest which can be represented as follow. point may occur. It is evident that the function does not liminf cos( n)1= − result in a negative value since each term is non-negative. n→∞ Therefore, it is safe to assume the lowest possible value is It suffices to say that since the value of the "n" and zero. On the other side, we propose that with the correct distance between two points are always non-negative values choice of n values we always would get zeros from the we can securely assume their multiplications would be at function in a periodic manner that will be explained in the least zero. That is in the subsequent sections of the existing proceeding sections. paper we would not investigate if negative values are possible solutions to the Littlewood conjecture. 3. Proof of Littlewood Conjecture 2. Preliminaries Lemma 1. ∀αβ, ∈ , α ∨∈ β ,lim inf nn||α |||| n β ||= 0 , Littlewood Conjecture states that n→∞ where for φ ∈ , we denote ||φ || as the distance to the liminfn→∞ nn||αβ ||.|| n ||= 0 , where ||b|| for b∈ nearest integer. Proof. To provide the proof, we need to separate the * Corresponding author: [email protected] (Kaveh Mozafari) proof into four main cases. Without loss of generality Published online at http://journal.sapub.org/ajms assume α is rational. Therefore, α can be write as Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved follow. 114 Kaveh Mozafari: Generalization and Proof of the Littlewood Conjecture p α =:,pq ∈≠ , q 0 q Where is the set of all integers. Case 1. α ≥ 0 Now we try to write α as mixed fraction. tt1 α =r: rtq , , ∈ , q ≠ 0,0 ≤< qq2 Note in such case the closest integer to α would be r t t tt ||α ||= ||r || = || r + || = | rr −− | = q q qq Also Note tt liminfnn ||αβ ||.|| n ||= lim inf nnr || ||.|| nβ ||= lim inf nnrn ||+ ||.|| nβ || nn→∞ →∞ qq n→∞ tt =liminfn | nr −− nr n|.|| nββ || =lim infn || n ||.|| n || nn→∞ qq→∞ Without loss of generality take: n∈≤,: n kq k ∈ tt liminfn || n ||.|| nβ ||= lim infkq || kq ||.|| kq ββ ||= lim infkq || kt ||.|| kq || nk→∞ qq→∞ k→∞ Note kt∈⇒ || kt || = | kt − kt | = 0 and kqt,,∈ we need to prove for any > 0 there exist an A > 0 such for all kA> the | (kq || kt ||.|| kqβ ||)−< 0 | holds. Since ‖|kt || is an integer then ||kt ||= 0 the following can be concluded. | (kq || kt ||.|| kqββ ||)−= 0 | | ( kq .0.|| kq ||)−=−= 0 | | 0 0 | 0 and 0<∴ lim infkq || kq− kt ||.|| kqβ || =0 k→∞ Therefore, for the presented case the proceeding statement is coorect. liminfnn ||αβ ||.|| n || = 0 n→∞ Case 2. α ≥ 0 Now we try to write α in mixed fraction format. tt1 α =r: rtq , , ∈ , q ≠ 0, ≤< 1 qq2 In the presented case, the closest integer to α would be r+1 t t t t qt− ||α ||= ||r || =+ || r || =+−−=−= | rr 1 | |1 | q q q qq Also Note tt liminfnn ||αβ ||.|| n ||= lim inf nnr || ||.|| nβ ||= lim inf nnrn ||+ ||.|| nβ || nn→∞ →∞ qq n→∞ tt =liminfn | n ( r +−− 1) nr n |.|| nββ || =lim infn | nr +−− n nr n|.|| n || nn→∞ qq→∞ t =liminfnnn || − ||.|| nβ || n→∞ q American Journal of Mathematics and Statistics 2017, 7(3): 113-124 115 Without loss of generality take: n∈≤,: n kq k ∈ tt liminfn || n−= n ||.|| nβ ||lim infkq || kq − kq ||.|| kq ββ || =lim infkq || kq − kt ||.||| kq | nk→∞ qq→∞ k→∞ Note kt∈⇒ kq −∈ kt,|| kt || = | kq −− kt ( kq − kt ) | = 0 and kqt,,∈ we need to prove for any > 0 there exist an A > 0 such for all kA> the | (kq || kq− kt ||.|| kqβ ||) −< 0 | holds. Since ‖|kq− kt || is an integer then ||kq−= kt || 0 the following can be concluded. | (kq || kq− kt ||.|| kqββ ||) −= 0 | | ( kq .0.|| kq ||)−=−= 0 | | 0 0 | 0 and 0<∴ lim infkq || kq− kt ||.|| kqβ || =0 k→∞ Therefore, for the presented case the proceeding statement is coorect. liminfnn ||αβ ||.|| n || = 0 n→∞ Case 3. α < 0 Now we try to write α in mixed fraction. tt1 α =−(r ) : rtq , , ∈ , q ≠ 0, 0 ≤< qq2 Note in such case the closest integer to α would be -r t t t t tt ||α ||=− || (r ) || =− || ( r + ) || =−−− | r ( ( r + )) | =−+ | rr + | = | | = q q q q qq Also Note tt liminfn || nαβ ||.|| n ||=lim infn || − nr ||.|| nβ || =lim inf n|| −− nr n ||.|| nβ || nn→∞ →∞ qq n→∞ tt =liminfn | −− nr ( nr − n ) |.|| nββ || =lim infn || n ||.|| n || nn→∞ qq→∞ Without loss of generality take: n∈≤,: n kq k ∈ tt liminfn || n ||.|| nβ |= lim inf kq || kq ||.|| kq ββ ||= lim infkq || kt ||.|| kq || n→∞ qqkq→∞ kq→∞ Note kt∈⇒ || kt || = | kt − kt | = 0 and kqt,,∈ we need to prove for any > 0 there exist an A > 0 such for all kA> the | (kq || kt ||.|| kqβ ||)−< 0 | holds. Since ‖|kt || is an integer then ||kt ||= 0 the following can be concluded. | (kq || kt ||.|| kqββ ||)−= 0 | | ( kq .0.|| kq ||)−=−= 0 | | 0 0 | 0 and 0<∴ lim infkq || kt ||.|| kqβ ||0= k→∞ Therefore, for the presented case the proceeding statement is coorect. liminfnn ||αβ ||.|| n || = 0 n→∞ Case 4. α < 0 Now we try to write α in mixed fraction. tt1 α =−(r ) : rtq , , ∈ , q ≠ 0, ≤< 1 qq2 Note in such case the closest integer to α would be -(r+1) 116 Kaveh Mozafari: Generalization and Proof of the Littlewood Conjecture t t t t tt ||α ||=− || (r ) || =−+ || ( r ) || =−+−−+ | ( r 1) ( ( r )) | =−−++ | rr 1 | =−+=− | 1 | 1 q q q q qq Also Note tt liminfn || nαβ ||.|| n ||=lim infn || − nr ||.|| nβ ||=lim infn || − nr − n ++ nr + n||.|| nβ || nn→∞ →∞ qq n→∞ t =liminfn || −+ nn ||.|| nβ || n→∞ q Without loss of generality take: n∈≤,: n kq k ∈ tt liminfn ||−+ n n ||.|| nβ || =lim inf kq|| −+ kq kq ||.|| kq ββ || =lim infkq || −+ kq kt ||| .| kq || n→∞ qqkq→∞ kq→∞ Note kt, kq∈⇒−+ | kq kt |∈⇒−+ || kq kt || = 0 and kqt,,∈ we need to prove for any > 0 there exist an A > 0 such for all kA> the | (kq ||− kq + kt ||.|| kqβ ||) −< 0 | holds. Since ‖|−+kq kt || is an integer then ||−+kq kt || = 0 the following can be concluded. | (kq ||−+ kq kt ||.|| kqββ ||) −= 0 | | ( kq .0.|| kq ||)−=−= 0 | | 0 0 | 0 and 0<∴ lim infkq || − kq + kt ||.|| kqβ || = 0 k→∞ Therefore, for the presented case the proceeding statement is coorect. liminfnn ||αβ ||.|| n || = 0 n→∞ That is, we have shown the Littlewood conjecture holds for all rational number. Corollary 1.1. Given αα12, ,..., αp∈ ,p ∈ , p ≠ 0, ∃α ii : i ∈ ,1 ≤≤ ip ,α ∈ , then we can conclude that liminfnn ||αα12 |||| n ||...|| n αp | |= 0 n→∞ ‖ By Lemma 1. we know the |nαi || would be equal to zero. We need to prove for any > 0 there exist an A > 0 ‖ such for all nA> the | (nn ||αα12 |||| n ||...|| n αip ||...|| n α ||)−< 0 | holds. Since |nαi || is an integer then ||||nαi ||||= 0 the following can be concluded. | (nn ||αα12 |||| n ||...|| n αip ||...|| n α ||)−= 0 | | ( nn || αα12 |||| n ||...0...|| n αp ||)−=−=−= 0 | | 0 0 | | 0 0 | 0 and 0<∴ lim infnn ||αα12 |||| n ||...| (| n αp |)0| = n→∞ Hence, we can conclude the following for the case that at least one of αα12, ,..., αp is a rational number. liminfnn ||αα12 |||| n ||...|| n αip ||...|| n α| | = 0 n→∞ Lemma 2. ∀∈αβ, ,lim infnn ||α |||| n β ||= 0 , where for φ ∈ , we denote ||φφ ||=minn∈ || − n ||.
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