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

Open access


In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.

[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 dilatations of integer sequences, II, in: Number Theory in Progress, Vol. 2 (K. Gy˝ory et al. eds.) Walter de Gruyter, Berlin-New York, 1999), pp. 633-638.

[5] CASSAIGNE, J.-MAUDUIT, C.-SÁRKÖZY, A.: On finite pseudorandom binary sequences VII: The measures of pseudorandomness, Acta Arith. 103 (2002), 97-118.

[6] CHERNOFF, H.: A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Stat. 23 (1952), 493-507.

[7] ERDŐS, P.-SÁRKÖZY, A.: Some solved and unsolved problems in combinatorial number theory, Math. Slovaca 28 (1978), 407-421.

[8] GYARMATI, K.: Measures of pseudorandomness, in: Finite fields and their Applications, (P. Charpin et al. eds.) Radon Ser. Comput. and Appl., De Gruyter, 2013, pp. 43-64.

[9] KOHAYAKAWA, Y.- MAUDUIT, C.- MOREIRA, C. G. - RÖDL, V.: Measures of pseudorandomness for finite sequences: minimum and typical values, in: Proc. of WORDS ’03, TUCS Gen. Publ.,Vol. 27, Turku Cent. Comput. Sci., Turku, 2003, 159-169.

[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 concerning irregularities of distribution of sequences of integers in arithmetic progressions. IV, Acta Math. Acad. Sci. Hungar. 30 no. 1-2 (1977), 155-162.

[15] SÁRKÖZY, A.-STEWART, C. L.: Irregularities of sequences relative to long arithmetic progressions, in: Analytic Number Theory, Essays in Honour of Roth, (W. W. R. Chen et al. eds.), Cambridge Univ. Press, Cambridge, (2009), 389-401.

[16] VALKÓ, B.: On irregularities of sums of integers, Acta Arith. 92 (2000), 367-381.

[17] VALKÓ, B.: Discrepancy of arithmetic progressions in higher dimensions, J. Number Theory 92 (2002), 117-130.

[18] Chernoff bound, Wikipedia, http://en.wikipedia.org/wiki/Chernoff-bound

Uniform distribution theory

The Journal of Slovak Academy of Sciences

Journal Information

Mathematical Citation Quotient (MCQ) 2017: 0.30


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 115 115 3
PDF Downloads 58 58 2