ISRN Combinatorics

Volume 2013 (2013), Article ID 979487, 4 pages

http://dx.doi.org/10.1155/2013/979487

Research Article

Partitions of Natural Numbers with the Intersection Not Empty

Wen Yu and Min Tang

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China

Received 14 May 2012; Accepted 19 July 2012

Academic Editors: A. Cossidente, M. Eliasi, L. Feng, and M. Gionfriddo

Copyright © 2013 Wen Yu and Min Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be the set of nonnegative integers. For a given set the representation functions , are defined as the number of solutions of the equation with condition , , respectively. In this paper, we prove that if and , then cannot hold for all sufficiently large integers where .

1. Introduction

Let be the set of nonnegative integers. For , let , , denote the number of solutions of

respectively. Sárközy asked ever whether there exist two sets and of nonnegative integers with infinite symmetric difference, that is:

for . As Dombi [1] has shown, the answer is negative for by the simple observation that is odd if and only if for some . For , Dombi presented a partition of the set of all positive integers into two disjoint subsets and such that for all sufficiently large . For , Chen and Wang [2] constructed sets such that for all integers , where is the set of all those positive integers such that the number of zeroes in the binary representation of is even, and is the set of all those positive integers such that the number of zeroes in the binary representation of is odd. For the other related results, the reader is referred to see [3–7].

It is natural to ask the following: for , are there subsets with for all large enough integers such that and ? Recently, the authors of this paper [8] obtained some results in this direction.

In this paper, we obtain the following results.

Theorem 1. If and , then cannot hold for all sufficiently large integers .

Theorem 2. If and , then cannot hold for all sufficiently large integers .

Remark 3. For , similarly, we can prove that if and , then cannot hold for all sufficiently large integers , where .

2. Proof of Theorem 1

Proof. Suppose that there exist integer and sets with and such that for all . Without loss of generality, we may assume that , is a positive integer; then there exists a polynomial of degree at most such that

Let

Then we have

Moreover,

Therefore,

Write

By (3) and (7), we have

Writing both sides of (9) in power series, then we have

Comparing the coefficients of on the both sides of (10), we have

Then we have

Since , we have . Then, by (12), has the following 30 cases:

By (13), we have

or

Thus has the following 10 cases:

By (14) and (25), we have which has the following 8 cases:

By (15) and (26), we have which has the following 6 cases:

By (16) and (27), we have which has the following 6 cases:

By (17) and (28), we have which has the following 2 cases:

By (18) and (29), we have which has the following 2 cases:

By (19) and (30), we have which has the following 2 cases:

By (20) and (31), we have which has the following 2 cases:

By (21), we have

or

which contradicts (32).

This completes the proof of Theorem 1.

3. Proof of Theorem 2

Proof. Suppose that there exist integer and sets with and such that for all . Without loss of generality, we may assume that , is a positive integer then there exists a polynomial of degree at most such that

Define , and as in the proof of Theorem 1. Then we have

Therefore,

Write

By (35) and (37), we have

We write both sides of (39) in power series; then we have

Comparing the coefficients of on the both sides of (40), we have

The remainder of the proof is the same as that of the proof of Theorem 1. We omit it here.

Acknowledgments

This work was supported by the National Natural Science Foundation of China, Grant no. 10901002 and Anhui Provincial Natural Science Foundation, Grant no. 1208085QA02.

References

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Y. G. Chen and B. Wang, “On additive properties of two special sequences,” Acta Arithmetica, vol. 110, no. 3, pp. 299–303, 2003.

Y. G. Chen, “On the values of representation functions,” Science China Mathematics, vol. 54, no. 7, pp. 1317–1331, 2011.

Y. G. Chen and M. Tang, “Partitions of natural numbers with the same representation functions,” Journal of Number Theory, vol. 129, no. 11, pp. 2689–2695, 2009.

V. F. Lev, “Reconstructing integer sets from their representation functions,” Electronic Journal of Combinatorics, vol. 11, article R78, 2004.

C. Sándor, “Partitions of natural numbers and their representation functions,” Integers, vol. 4, article A18, 2004.

M. Tang, “Partitions of the set of natural numbers and their representation functions,” Discrete Mathematics, vol. 308, pp. 2614–2616, 2008.

W. Yu and M. Tang, “A note on partitions of natural numbers and their representation functions,” preprint