Ran Ji and Craig V. Spencer

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 19, 2013, Number 3, Pages 55—59

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## Details

### Authors and affiliations

Ran Ji

*Department of Mathematics, Wellesley College
106 Central Street, Wellesley, MA 02481, USA*

Craig V. Spencer

*Department of Mathematics, Kansas State University
138 Cardwell Hall, Manhattan, KS 66506, USA*

### Abstract

Let D(G) be the maximal cardinality of a set *A* ⊆ *G* that contains no non-trivial solution to *x*_{1} + … + *x _{s}* −

*sx*

_{s+1}= 0 with

*x*∈

_{i}*A*(1 ≤

*i*≤

*s*+ 1). Let

where

*rk*(

*H*) is the rank of

*H*. We prove that for any

*n*∈ ℕ, , where is a fixed constant depending only on

*s*.

### Keywords

- Finite Abelian groups
- Character sums

### AMS Classification

- 11B30
- 20D60
- 11T24

### References

- Lev, V. F. Progression-free sets in finite abelian groups, J. Number Theory Vol. 104, 2004, 162–169.
- Liu, Y.-R., C. V. Spencer, A generalization of Meshulam’s Theorem on subsets of finite abelian groups with no 3-term arithmetic progression, Design. Code. Cryptogr., Vol. 52, 2009, 83–91.
- Meshulam, R. On subsets of finite abelian groups with no 3-term arithmetic progressions, J. Combin. Theory Ser. A, Vol. 71, 1995, 168–172.
- Serre, J.-P. A Course in Arithmetic, Springer-Verlag, New York, 1973.

## Related papers

## Cite this paper

APAJi, R., & Spencer, C. V. (2013). On subsets of finite Abelian groups without non-trivial solutions of *x*_{1 }+ *x*_{2} + … + *x*_{s} –* sx _{s}*

_{+1}= 0. Notes on Number Theory and Discrete Mathematics, 19(3), 55-59.

Ji, Ran, and Craig V. Spencer. “On Subsets of Finite Abelian Groups without Non-trivial Solutions of *x*_{1 }+ *x*_{2} + … + *x*_{s} –* sx _{s}*

_{+1}= 0.” Notes on Number Theory and Discrete Mathematics 19, no. 3 (2013): 55-59.

Ji, Ran, and Craig V. Spencer. “On Subsets of Finite Abelian Groups without Non-trivial Solutions of *x*_{1 }+ *x*_{2} + … + *x*_{s} –* sx _{s}*

_{+1}= 0.” Notes on Number Theory and Discrete Mathematics 19.3 (2013): 55-59. Print.