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On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, I. (General Results)

-169. [10] MATOUŠEK, J.-SPENCER, J.: Discrepancy in arithmetic progression, J. Amer.Math. Soc. 9 (1996), 195-204. [11] MAUDUIT, C.- SÁRKÖZY, A.: On finite pseudorandom binary sequences, I. Measure of pseudorandomness, the Legendre symbol, Acta Arith. 82 (1997), 365-377. [12] MÉRAI, L.: The higher dimensional analogue of certain estimates of Roth and S´ark¨ozy, Period. Math. Hung. (to appear) [13] ROTH, K. F. : Remark concerning integer sequences, Acta Arith. 9 (1964), 257-260. [14] SÁRKÖZY, A.: Some remarks

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On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

REFERENCES [1] BURGESS, D. A.: On character sums and primitive roots , Proc. London Math. Soc. 1 2 (1962), no. 3, 179–192. [2] DARTYGE, C.—GYARMATI, K.—SÁRKÖZY, A.: On irregularities of distribution of binary sequences relative to arithmetic progressions, I. (General results) , Unif. Distrib. Theory 1 2 (2017), no. 1, 55–67. [3] DAVENPORT, H.—ERDŐS, P.: The distribution of quadratic and higher residues , Publ. Math. Debrecen 2 (1952), 252–265. [4] ERDŐS, P.—SÁRKÖZY, A.: Some solved and unsolved problems in combinatorial number

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On Super (a, d)-H-Antimagic Total Covering of Star Related Graphs

Abstract

Let G = (V (G),E(G)) be a simple graph and H be a subgraph of G. G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection λ: V (G) ∪ E(G) → {1, 2, 3, . . . , |V (G)| + |E(G)|} such that for all subgraphs H′ isomorphic to H, the H′ weights

constitute an arithmetic progression a, a+d, a+2d, . . . , a+(n−1)d where a and d are positive integers and n is the number of subgraphs of G isomorphic to H. Additionally, the labeling λ is called a super (a, d)-H-antimagic total labeling if λ(V (G)) = {1, 2, 3, . . . , |V (G)|}.

In this paper we study super (a, d)-H-antimagic total labelings of star related graphs Gu[Sn] and caterpillars.

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Uncanny Subsequence Selections That Generate Normal Numbers

Abstract

Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)

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Motzkin’s Maximal Density and Related Chromatic Numbers

channel assignment problem for mutually adjacent sites , J. Combin. Theory Ser. A 68 (1994), 169–183. [6] GUPTA, S.—TRIPATHI, A.: Density of M-sets in arithmetic progression Acta Arith. 89 (1999), 255–257. [7] HARALAMBIS, N. M.: Sets of integers with missing differences , J. Combin. Theory Ser. A 23 (1977), 22–33. [8] LIU, D. D.-F.—ZHU, X.: Fractional chromatic number for distance graphs with large clique size , J. Graph Theory 47 (2004), 129–146. [9] LIU, D. D.-F.—ZHU, X.: Fractional chromatic number of distance graphs generated by two

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Uniform Distribution with Respect to Density

. Theory 2 (2007), no. 1, 79-92. PAŠTÉKA, M.-ŠAL´AT, T.: Buck’s measure density and sets of positive integers containing arithmetic progressions, Math. Slovaca 41 (1991), 283-293. PAŠTÉKA, M.-PORUBSK´Y, V S.: On the distribution of sequences of integers, Math. Slovaca 43 (1993), 521-639. PAŠTÉKA, M.-TICHY, R. F.: Distribution Problems in Dedekind Domains and Submeasures, Annali dell’ Universita di Ferrara, Sezione VII-Scienze Matematiche 40 (1994), 191-206. STRAUCH, O-PORUBSK´Y, Š.: Distribution of Sequences

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Questions Around the Thue-Morse Sequence

’entiers , Ann. Inst. Fourier (Grenoble) 55 (2005), 2423–2474. [10] DOCHE, C.—HABSIEGER, L.: Moments of the Rudin-Shapiro polynomials , J. Fourier Anal. Appl. 10 (2004), 497–505. [11] DUMONT, J. M.: Discrépance des progressions arithmétiques dans la suite de Morse , C. R. Acad. Sci. Paris, SZ̆r. I 297 (1983), 145–148. [12] FALCONER, K.: Fractal geometry, mathematical foundations and applications , John Wiley & sons, 1990. [13] FOUVRY, E.—MAUDUIT, C.: Sommes des chiffres et nombres presque premiers , Math. Ann. 305 (1996), 571–599. [14] GELFOND

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