Geodesic
Partitions of \(\mathbb Z_m\) with the same weighted representation functions
Zhenhua Qu1
1East China Normal University
The electronic journal of combinatorics, Tome 21 (2014) no. 2
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Let $\mathbf{k}=(k_1,k_2,\cdots,k_t)$ be a $t$-tuple of integers, and $m$ be a positive integer. For a subset $A\subset\mathbf{Z}_m$ and any $n\in\mathbf{Z}_m$, let $r_A^{\mathbf{k}}(n)$ denote the number of solutions of the equation $k_1a_1+\cdots+k_ta_t=n$ with $a_1,\cdots,a_t\in A$. In this paper, we give a necessary and sufficient condition on $(\mathbf{k},m)$ such that there exists a subset $A\subset \mathbf{Z}_m$ satisifying $r_{A}^{\mathbf{k}}=r_{\mathbf{Z}_m\backslash A}^{\mathbf{k}}$. This settles a problem of Yang and Chen.
DOI : 10.37236/4089
Classification : 11B34, 05A15, 05A17
Mots-clés : representation function, partition, Sárközy problem

Zhenhua Qu  1

1 East China Normal University 
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     author = {Zhenhua Qu},
     title = {Partitions of \(\mathbb {Z_m\)} with the same weighted representation functions},
     journal = {The electronic journal of combinatorics},
     year = {2014},
     volume = {21},
     number = {2},
     doi = {10.37236/4089},
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Zhenhua Qu. Partitions of \(\mathbb Z_m\) with the same weighted representation functions. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/4089

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