On sidon sequences of farey sequences, square roots and reciprocals
-
Gergő Surányi
Abstract
In this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that Sidon sequences over them have cardinality of between
Communicated by Milan Paštéka
References
[1] Babai, L.—Sós, V. T.: Sidon sets in groups and induced subgraphs of Cayley graphs, European J. Combin. 6 (1985), 101–114.10.1016/S0195-6698(85)80001-9Suche in Google Scholar
[2] Cilleruelo, J.: Infinite Sidon sequences, Adv. Math. 255 (2014), 474–486.10.1016/j.aim.2014.01.011Suche in Google Scholar
[3] Cilleruelo, J.—Kiss, S.—Ruzsa, I. Z.—Vinuesa, C.: Generalization of a theorem of Erdős and Rényi on Sidon sequences, Random Structures Algorithms 37 (2010), 455–464.10.1002/rsa.20350Suche in Google Scholar
[4] Cilleruelo, J.—Ruzsa, I.—Vinuesa, C.: Generalized Sidon sets, Adv. Math. 225 (2010), 2786–280710.1016/j.aim.2010.05.010Suche in Google Scholar
[5] Deshouillers, J.-M.—Plagne, A.: A Sidon basis, Acta Math. Hungar. 123 (2009), 233–238.10.1007/s10474-008-8097-3Suche in Google Scholar
[6] Erdős, P.: On a problem of Sidon in additive number theory and on some related problems. Addendum, J. London Math. Soc. 19 (1944), 208.10.1112/jlms/19.76_Part_4.208Suche in Google Scholar
[7] Erdős, P.—Freud, R.: On Sidon sequences and related problems, Mat. Lapok 1 (1991), 1–44 (in Hungarian).Suche in Google Scholar
[8] Erdős, P.—Sárközy, A.: Problems and results on additive properties of general sequences II, Acta Math. Hung. 48 (1986), 201–211.10.1007/BF01949065Suche in Google Scholar
[9] Erdős, P.—Surányi, J.: Válogatott fejezetek a számelméletből, POLYGON Szeged (2004), 238–239 (in Hungarian).Suche in Google Scholar
[10] Erdős, P.—Turán, P.: On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc. 16 (1941), 212–215.10.1112/jlms/s1-16.4.212Suche in Google Scholar
[11] Graham, R. L. et al. (eds.): The Mathematics of Paul Erdős I, 2013, pp. 233–262.10.1007/978-1-4614-7254-4Suche in Google Scholar
[12] Graham, R. L.—Sloane, N. J. A.: On additive bases and harmonious graphs, SIAM J. Algebraic Discrete Methods 1 (1980), 382-404. MR 82f:10067a10.1137/0601045Suche in Google Scholar
[13] Graham, R. L.—Sloane, N. J. A.: On constant weight codes and harmonious graphs. In: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Utilitas Math., Winnipeg, Man., 1980, pp. 25–40.Suche in Google Scholar
[14] Gyárfás, A.—Lehel, Z.: Linear sets with five distinct differences among any four elements, J. Combin. Theory Ser. B64 (1995), 108–118.10.1006/jctb.1995.1028Suche in Google Scholar
[15] Halberstam, H.—Roth, K.: Sequences, rev. ed., New York:Springer-Verlag, 1983, pp. 84–9710.1007/978-1-4613-8227-0Suche in Google Scholar
[16] Kiss, S. Z.: On Sidon sets which are asymptotic bases, Acta Math. Hungar. 128 (2010), 46–58.10.1007/s10474-010-9155-1Suche in Google Scholar
[17] Komlós, J.—Sulyok, M.—Szemerédi, E.: A lemma of combinatorial number theory, Mat. Lapok 23 (1972), 103–108 (in Hungarian, with english summary).Suche in Google Scholar
[18] Komlós, J.—Sulyok, M.—Szemerédi, E.: Linear problems in combinatorial number theory, Acta Math. Acad. Sci. Hungar. 26 (1975), 113–121.10.1007/BF01895954Suche in Google Scholar
[19] O’Bryant, K.: A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. 11:39 (2004).10.37236/32Suche in Google Scholar
[20] Ruzsa, I. Z.: Solving a linear equation in a set of integers. I, Acta Arith. 65 (1993), 259–282.10.4064/aa-65-3-259-282Suche in Google Scholar
[21] Ruzsa, I. Z.: A small maximal Sidon set, Ramanujan J. 2 (1998), 55–58.10.1007/978-1-4757-4507-8_6Suche in Google Scholar
[22] Ruzsa, I. Z.: An infinite Sidon sequence, J. Number Theory 68 (1998), 63–71. Article No. NT97219210.1006/jnth.1997.2192Suche in Google Scholar
[23] Ruzsa, I. Z.: Additive and multiplicative Sidon sets, Acta Math. Hungar. 112 (2006), 345–354.10.1007/s10474-006-0102-0Suche in Google Scholar
[24] Sós, V. T.: An additive problem on different structures. 3rd Internat. Comb. Conf., San Francisco 1989. Graph Theory, Comb. Alg. and Appl. SIAM, (Y. Alavi, F. R. K. Chung, R. L. Graham, D. F. Hsu, eds.), 1991, pp. 486–508.Suche in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Recurrences for the genus polynomials of linear sequences of graphs
- The existence of states on EQ-algebras
- On sidon sequences of farey sequences, square roots and reciprocals
- Repdigits as sums of three balancing numbers
- Lipschitz one sets modulo sets of measure zero
- Refinement of fejér inequality for convex and co-ordinated convex functions
- An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
- Coefficients problems for families of holomorphic functions related to hyperbola
- Triebel-Lizorkin capacity and hausdorff measure in metric spaces
- The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
- On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
- Density of summable subsequences of a sequence and its applications
- Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
- A note on cosine series with coefficients of generalized bounded variation
- Some geometric properties of the non-Newtonian sequence spaces lp(N)
- On sequence spaces defined by the domain of a regular tribonacci matrix
- Jointly separating maps between vector-valued function spaces
- Some fixed point theorems for multi-valued mappings in graphical metric spaces
- Monotone transformations on the cone of all positive semidefinite real matrices
- Locally defined operators in the space of Ck,ω-functions
- Some properties of D-weak operator topology
- Estimating the distribution of a stochastic sum of IID random variables
- An internal characterization of complete regularity
Artikel in diesem Heft
- Regular papers
- Recurrences for the genus polynomials of linear sequences of graphs
- The existence of states on EQ-algebras
- On sidon sequences of farey sequences, square roots and reciprocals
- Repdigits as sums of three balancing numbers
- Lipschitz one sets modulo sets of measure zero
- Refinement of fejér inequality for convex and co-ordinated convex functions
- An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
- Coefficients problems for families of holomorphic functions related to hyperbola
- Triebel-Lizorkin capacity and hausdorff measure in metric spaces
- The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
- On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
- Density of summable subsequences of a sequence and its applications
- Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
- A note on cosine series with coefficients of generalized bounded variation
- Some geometric properties of the non-Newtonian sequence spaces lp(N)
- On sequence spaces defined by the domain of a regular tribonacci matrix
- Jointly separating maps between vector-valued function spaces
- Some fixed point theorems for multi-valued mappings in graphical metric spaces
- Monotone transformations on the cone of all positive semidefinite real matrices
- Locally defined operators in the space of Ck,ω-functions
- Some properties of D-weak operator topology
- Estimating the distribution of a stochastic sum of IID random variables
- An internal characterization of complete regularity
