1. Introduction

Littlewood Conjecture is an interesting conjecture in mathematics which was proposed by John Edensor Littlewood close to 90 years ago. The conjecture considered an open problem in mathematics since 2016. The presented paper proves the conjecture while it generalizes it to a theorem which states that limit inferior of multiplication of a finite number of real number's distance from the nearest integer is always zero. By limit inferior, we are referring to the lower bound of the limit upon its existence. For example limit inferior of cosine function for real value would be -1, which can be represented as follow.

It suffices to say that since the value of the "n" and distance between two points are always non-negative values we can securely assume their multiplications would be at least zero. That is in the subsequent sections of the existing paper we would not investigate if negative values are possible solutions to the Littlewood conjecture.

2. Preliminaries

Littlewood Conjecture states that , where ||b|| for (set of all real numbers) means the distance of b from the closest integer while . Moreover is defined as follow.

In the present paper the Littlewood conjecture is proven to be correct by first proving the conjecture for rational number and then extending it to all real numbers. At the end the conjecture is extended to cover more terms as follow.

To get limit inferior, we need to realize where the lowest point may occur. It is evident that the function does not result in a negative value since each term is non-negative. Therefore, it is safe to assume the lowest possible value is zero. On the other side, we propose that with the correct choice of n values we always would get zeros from the function in a periodic manner that will be explained in the proceeding sections.

3. Proof of Littlewood Conjecture

Lemma 1. , where for , we denote as the distance to the nearest integer.

Proof. To provide the proof, we need to separate the proof into four main cases. Without loss of generality assume is rational. Therefore, can be write as follow.

Where is the set of all integers.

Case 1. Now we try to write as mixed fraction.

Note in such case the closest integer to would be r

Also Note

Without loss of generality take:

Note and we need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Therefore, for the presented case the proceeding statement is coorect.

Case 2. Now we try to write in mixed fraction format.

In the presented case, the closest integer to would be r+1

Also Note

Without loss of generality take:

Note and we need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Therefore, for the presented case the proceeding statement is coorect.

Case 3. Now we try to write in mixed fraction.

Note in such case the closest integer to would be -r

Also Note

Without loss of generality take:

Note and we need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Therefore, for the presented case the proceeding statement is coorect.

Case 4. Now we try to write in mixed fraction.

Note in such case the closest integer to would be -(r+1)

Also Note

Without loss of generality take:

Note and we need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Therefore, for the presented case the proceeding statement is coorect.

That is, we have shown the Littlewood conjecture holds for all rational number.

Corollary 1.1. Given then we can conclude that

By Lemma 1. we know the would be equal to zero. We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Hence, we can conclude the following for the case that at least one of is a rational number.

Lemma 2. , where for , we denote

Proof. To provide the proof, we need to the matters into four main cases. Without loss of generality we proceed with proof with focusing on an arbitrary the irrational number .

Case 5.

We can express as follow.

Where is the floor function.

Note:

The closest integer to with the given constrains would be

Without loss of generality assume

By Archimedes Axiom

Where

Assign Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Now take

Also Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Note

By Squeeze theorem

From

Case 6.

We can express as follow.

Note

The closest integer to with the given constrains would be

Without loss of generality assume

By Archimedes Axiom

Where

Assign Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Now take

Also Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Note:

By Squeeze theorem

Since

Case 7.

We can express as follow.

Where is the floor function.

Note:

The closest integer to with the given constrains would be

Without loss of generality take

By Archimedes Axiom

Where

Assign

Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Take

Also Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Note:

By Squeeze theorem

From

Case 8.

We can express as follow.

Note

The closest integer to with the given constrains would be

Without loss of generality

By Archimedes Axiom

Where

Assign

Note

Without loss of generality

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Now take

Also Note

Without loss of generality take:

Note . We need to prove for any there exist an such for all the holds. Since is an integer then the following can be concluded.

and

Note:

By Squeeze theorem

Since

In all cases we have concluded the following.

Therefore we can say the Little Conjecture holds for all real numbers.

Corollary 2.1. Given , Then we can conclude that

Without loss of generality, by Lemma 2 we have the following.

By Squeeze theorem

4. Generalization

It suffices to focus on the generalized form of Littlewood conjecture based on the provided proof. Consequently, the following theorem is expectedly obtained.

Theorem 3. , where for , we denote

Proof. To prove the theorem we refer to the Corollary 1.1. and Corollary 2.1. which are resulted from Lemma 1. and Lemma 2. respectively. The two corollary contain all possibilities of over the real numbers. Therefore we can conclude.

The possible applications of the theorem are beyond the scope of the presented paper, and hopefully, subsequent papers would cover portions of such claim.