A note on sumsets and difference sets in groups of order 12

Neetu; Anupriya  Shetty; B R Shankar

doi:10.5269/bspm.78446

Neetu National Institute of Technology Karnataka, Surathkal
Anupriya Shetty Department of Mathematics, St Aloysius (Deemed to be University), Mangaluru, Karnataka, 575003, India
B R Shankar National Institute of Technology Karnataka, Surathkal

Resumo

A subset $A$ of a group $G$ is referred to as a balanced set when $|A+A|=|A-A|$, MSTD (more sums than differences) when $|A+A|>|A-A|$, and MDTS (more differences than sums) when $|A-A|>|A+A|$. In this paper, we present a comparative study of MSTD and MDTS sets in groups of order 12 up to isomorphism. Additionally, we have completely categorized such sets in these groups and have provided a set $A$ with the current highest value of $ln(|A+A|)/ln(|A-A|).$

Downloads

Não há dados estatísticos.

Referências

Asada, M., Manski, S., Miller, S. and Suh, H., Fringe pairs in generalized sum-dominant sets, Int. J. Number Theory 13, 2653–2675, (2017).

Ascoli, R., Cheigh, J., Dantas e Moura, G. Z., Jeong, R., Keisling, A., Lilly, A., Miller, S., Ngamlamai, P. and Phang, M., Sum and difference sets in generalized dihedral groups, arXiv preprint, arXiv:2210.00669, (2022).

Chu, H., Luntzlara, N., Miller, S. and Shao, L., Generalizations of a curious family of sum-dominant sets hidden by interior blocks, Integers 20A, (2020).

Chu, H., King, D., Luntzlara, N., Martinez, T., Miller, S., Shao, L., Sun, C. and Xu, V., Generalizing the distribution of missing sums in sumsets, J. Number Theory 239, 402–444, (2022).

Do, T., Kulkarni, A., Miller, S., Moon, D. and Wellens, J., Sums and differences of correlated random sets, J. Number Theory 147, 44–68, (2015).

Freiman, G. and Pigarev, V., The relation between the invariants R and T, Kalinin. Gos. Univ. Moscow (Russian), 172–174, (1973).

Hegarty, P., Some explicit constructions of sets with more sums than differences, Acta Arith. 130, 61–77, (2007).

Hemmady, A., Lott, A. and Miller, S., When almost all sets are difference-dominated in Zn, Integers 15, A54, (2020).

Marica, J., On a conjecture of Conway, Canad. Math. Bull. 12, 233–234, (1969).

Miller, S. and Vissuet, K., Most subsets are balanced in finite groups, in Combinatorial and Additive Number Theory, 101, 147–157, (2014).

Nathanson, M., Sets with more sums than differences, Integers: Electron. J. Combin. Number Theory 7, A5 (24pp), (2007).

Neetu and Shankar, B. R., Sum and difference sets in generalized quaternion groups, Proc. Jangjeon Math. Soc. 27, no. 4, 773–780, (2024).

Penman, D. and Wells, M., On sets with more restricted sums than differences, Integers 13, A57, (2013).

Penman, D. and Wells, M., Sum-dominant sets and restricted-sum-dominant sets in finite abelian groups, Acta Arith. 165, 361–383, (2014).

Zdunczyk, R., Sets with small differences and big sums in finite groups, in Selected Problems on Experimental Mathematics, 672, 149–166, (2017).

Zhao, Y., Counting sum-dominant sets in finite abelian groups, J. Number Theory 130, 2308–2322, (2010).