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References [1] KITAOKA,Y.: Statistical distribution of roots of a polynomial modulo primes, (submitted). [2] The On-Line Encyclopedia of Integer Sequences, Published electronically, 2000, http://oeis.org [3] SERRE, J. P.: Quelques applications du théorème de densité de Chebotarev, I.H.E.S., 54 (1981), 323-401.

References BINDER, C.: Über einen Satz von de Bruijn und Post, ¨Osterr. Akad. Wiss. Math.-Naturw. Kl. SB. II, 179 (1971), 233-251. DE BRUIJN, N.G.-POST, K. A.: A remark on uniformly distributed sequences and Riemann integrability Indag. Math. 30 (1968), 149-150. BUCK, R. C.: The measure theoretic approach to density, Amer. J. Math 68 (1946), 560-580. CRISTEA, L. L.-PRODINGER, H.: Moments of distributions related to digital expansions, J. Math. Anal. Appl. 315 (2006), 606-625. CRISTEA, L. L.-TICHY, R. F.: Discrepancies of point sequences on the Sierpi´nski carpet

REFERENCES [FT] FILIP, F.—TÓTH, J. T.: Characterization of asymptotic distribution functions of ratio block sequences , Periodica Mathematica Hungarica 60 (2010), no. 2, 115–126. 10.1007/s10998-010-2115-2 [GS] G: REKOS, G.—STRAUCH, O. Distribution functions of ratio sequences, II , Unif. Distrib. Theory 2 (2007), no. 1, 53–77. [GV] GREKOS, G.—VOLKMANN, B.: On densities and gaps , Journal of Number Theory 26 (1987), 129–148. 10.1016/0022-314X(87)90074-6 [Po] PÓLYA, G.: Untersuchungen über Lücken und Singularitäten von Potenzreihen , Math. Zeit. 29

REFERENCES [BW] BUCKLEW, J. A. — WISE, G. L.: Multidimensional asymptotic quantization theory with rth power distortion measures , IEEE Trans. Inform. Theory, 28 (1982), no. 2, 239–247. [CR]ÇÖMEZ, D.—ROYCHOWDHURY, M. K.: Quantization for uniform distributions on stretched Sierpiński triangles , Monatsh. Math. 190 (2019), no. 1, 79–100. [DR1] DETTMANN, C. P.—ROYCHOWDHURY, M. K.: Quantization for uniform distributions on equilateral triangles , Real Anal. Exchange, 42 (2017), no. (1), 149–166. [DR2] DETTMANN, C. P.—ROYCHOWDHURY, M. K.: An algorithm to

REFERENCES [1] BAKER, R. C.—HARMAN, G.—PINTZ, J.: The difference between consecutive primes, II , Proc. London Math. Soc. (3) 83 (2001), no. 3, 532–562. 10.1112/plms/83.3.532 [2] BALÁŽ, V.—NAGASAKA, K.—STRAUCH, O.: Benford’s law and distribution functions of sequences in (0, 1) (Russian), Mat. Zametki 88 (2010), no. 4, 485–501; (English translation) Math. Notes 88 (2010), no. 4, 449–463. [3] BENFORD, F.: The law of anomalous numbers , Proc. Amer. Phil. Soc. 78 (1938), 551–572 (Zbl 18, 265; JFM 64.0555.03). [4] DIACONIS, P.: The distribution of

REFERENCES [AW] ABAYA, E. F.—WISE, G. L.: Some remarks on the existence of optimal quantizers , Statistics & Probability Letters 2 (1984), no. 6, 349–351. [C] CHUNG, K.-L.: An estimate concerning the Kolmogoroff limits distribution , Trans. Amer. Math. Soc. 67 (1949), 36–50. [PC] COHORT, P.: Limit theorems for random normalized distortion , Ann. Appl. Probab. 14 (2004), no. 1, 118–143. [D] DEHEUVELS, P.: Strong bounds for multidimensional spacings , Z. Wahrsch. Verw. Gebiete 64 (1983, 411–424. [DR] DETTMANN, C. P.—ROYCHOWDHURY, M. K.: Quantization

REFERENCES [1] BARTLE, R. G.—SHERBERT, D. R.: Introduction to Real Analysis (4th eddition). John Wiley & Sons, Inc., New York, 2011. [2] BENFORD, F.: The law of anomalous numbers , Proc. Am. Philos. Soc., 78(4) (1938), 551–572. [3] JANG, D.—KANG, J. K.—KRUCKMAN, A.—KUDO, J.—MILLER, S. J.: Chains of distributions, hierarchical Bayesian models and Benford’s law , J.Algebra, Number Theory, Adv. Appl. 1 (2009), no. 1, 37–60. [4] KOSSOVSKY, A. E.: Benford’s Law. Theory, The General Law of Relative Quantities, And Forensic Fraud Detection Applications , World

References [1] AISTLEITNER, C.: On the limit distribution of the well-distribution measure of random binary sequences, J. Thor. Nombres Bordeaux 25 (2013), no. 2, 245-259. [2] ALON, N.-KOHAYAKAWA, Y.- MAUDUIT, C.-MOREIRA, C. G. - RÖDL, C. G.: Measures of pseudorandomness for finite sequences: typical values, Proc. Lond. Math. Soc. 95 (2007), 778-812. [3] BECK, J.: Roth’s estimate of the discrepancy of integer sequences is nearly sharp, Combinatorica 1 (1981), 319-325. [4] BECK, J.-SÁRKÖZY, A.-STEWART, V.: On irregularities of distribution in shifts and

of roots of a polynomial in different local fields, Math. Comput. 78 (2009), no. 265, 523-536. [4] A statistical relation of roots of a polynomial in different local fields II, (Aoki, Takashi ed. et al.) In: Number Theory: Dreaming in Dreams, Proceedings of The 5th China-Japan seminar, Higashi-Osaka, Japan, August 27-31, 2008. Ser. Number Theory Appl. Vol. 6, 2010, World Sci. Publ., Hackensack, NJ, pp. 106-126. [5] A statistical relation of roots of a polynomial in different local fields III, Osaka J. Math. 49 (2012), 393-420. [6] Statistical distribution of roots

REFERENCES [K1] KITAOKA, Y.: Notes on the distribution of roots modulo a prime of a polynomial, Unif. Distrib. Theory 12 (2017), no. 2, 91–116. [K2] –––––– Statistical distribution of roots of a polynomial modulo primes III ,Int. J.Statist. Probab. 7 (2018), 115–124.