###### 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

###### 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

###### 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 G_{u}[S_{n}] and caterpillars.

###### Uncanny Subsequence Selections That Generate Normal Numbers

## Abstract

Given a real number 0_{.a1a2a3} . . . that is normal to base b, we examine increasing sequences n_{i} so that the number 0_{.an1an2an3} . . . are normal to base b. Classically, it is known that if the n_{i} form an arithmetic progression, then this will work. We give several more constructions including n_{i} that are recursively defined based on the digits a_{i}. 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)

###### Motzkin’s Maximal Density and Related Chromatic Numbers

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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

###### 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