Czechoslovak Mathematical Journal, first online, pp. 1-8


Sidon basis in polynomial rings over finite fields

Wentang Kuo, Shuntaro Yamagishi

Received December 21, 2019.   Published online September 24, 2020.

Abstract:  Let $\mathbb{F}_q[t]$ denote the polynomial ring over $\mathbb{F}_q$, the finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is not $2$ or $3$. We prove that there exist infinitely many $N \in\mathbb{N}$ such that the set $\{ f \in\mathbb{F}_q[t] \colon\deg f < N \}$ contains a Sidon set which is an additive basis of order $3$.
Keywords:  Sidon set; additive basis; polynomial rings over finite fields
Classification MSC:  11K31, 11B83, 11T55
DOI:  10.21136/CMJ.2020.0543-19

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References:
[1] J. Cilleruelo: Combinatorial problems in finite fields and Sidon sets. Combinatorica 32 (2012), 497-511. DOI 10.1007/s00493-012-2819-4 | MR 3004806 | Zbl 1291.11025
[2] J. Cilleruelo: On Sidon sets and asymptotic bases. Proc. Lond. Math. Soc. (3) 111 (2015), 1206-1230. DOI 10.1112/plms/pdv050 | MR 3477233 | Zbl 1390.11026
[3] J.-M. Deshouillers, A. Plagne: A Sidon basis. Acta Math. Hung. 123 (2009), 233-238. DOI 10.1007/s10474-008-8097-3 | MR 2500912 | Zbl 1200.11008
[4] P. Erdős, A. Sárközy, V. T. Sós: On additive properties of general sequences. Discrete Math. 136 (1994), 75-99. DOI 10.1016/0012-365X(94)00108-U | MR 1313282 | Zbl 0818.11009
[5] P. Erdős, A. Sárközy, V. T. Sós: On sum sets of Sidon sets I. J. Number Theory 47 (1994), 329-347. DOI 10.1006/jnth.1994.1040 | MR 1278402 | Zbl 0811.11014
[6] P. Erdős, P. Turán: On a problem of Sidon in additive number theory, and on some related problems. J. Lond. Math. Soc. 16 (1941), 212-215. DOI 10.1112/jlms/s1-16.4.212 | MR 0006197 | Zbl 0061.07301
[7] S. Z. Kiss: On Sidon sets which are asymptotic basis. Acta Math. Hung. 128 (2010), 46-58. DOI 10.1007/s10474-010-9155-1 | MR 2665798 | Zbl 1218.11012
[8] S. Z. Kiss, E. Rozgonyi, C. Sándor: On Sidon sets which are asymptotic bases of order 4. Funct. Approximatio, Comment. Math. 51 (2014), 393-413. DOI 10.7169/facm/2014.51.2.10 | MR 3282635 | Zbl 1353.11016
[9] S. V. Konyagin, V. F. Lev: The Erdős-Turán problem in infinite groups. Additive Number Theory. Springer, New York (2010), 195-202. DOI 10.1007/978-0-387-68361-4_14 | MR 2744757 | Zbl 1271.11011
[10] S. Lang, A. Weil: Number of points of varieties in finite fields. Am. J. Math. 76 (1954), 819-827. DOI 10.2307/2372655 | MR 0065218 | Zbl 0058.27202
[11] K. O'Bryant: A complete annotated bibliography of work related to Sidon sequences. Electron. J. Comb. DS11 (2004), 39 pages. Zbl 1142.11312

Affiliations:   Wentang Kuo, Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada, e-mail: wtkuo@uwaterloo.ca; Shuntaro Yamagishi (corresponding author), Mathematical Institute, Utrecht University, Budapestlaan 6, 3584 CD Utrecht, The Netherlands, e-mail: s.yamagis@uu.nl


 
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