#A INTEGERS 6 (206) STRONG NORMALITY AND GENERALIZED COPELAND ERDŐS NUMBERS Elliot Catt School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia Elliot.Catt@uon.edu.au Michael Coons 2 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia Michael.Coons@newcastle.edu.au Jordan Velich 3 School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia Jordan.Velich@uon.edu.au Received: 3/5/5, Revised: 0/2/5, Accepted: /30/6, Pulished: 2/29/6 Astract We prove that an infinite class of Copeland-Erdős numers is not strongly normal and provide the analogous result for Bugeaud s Mahler-inspired extension of the Copeland-Erdős numers. After the presentation of our results, we o er several open questions concerning normality and strong normality.. Introduction Up to the year 909, prolems of proaility were classified as either discontinuous or continuous (also called geometric ). Towards filling this gap, in that year, Borel [2] introduced what he called countale proailities (proaités dénomrales). In this new type of prolem, one ass proailistic questions aout countale sets. As a (now very common) canonical example, Borel [2] considered properties of the frequency of digits in the digital expansions of real numers. Specifically, he defined the notions of simple normality and normality. Catt was supported y a CARMA Undergraduate Summer Research Scholarship. 2 Coons was supported y ARC grant DE4000223. 3 Velich was supported y a CARMA Undergraduate Summer Research Scholarship.
INTEGERS: 6 (206) 2 A real numer x is called simply normal to the ase (or -simply normal) if each of 0,,..., occurs in the ase- expansion of x with equal frequency /. This numer x is then called normal to the ase (or -normal) provided it is m -simply normal for all positive integers m. As part of his seminal wor in the area, Borel [2] showed that almost all numers are normal to every positive integer ase, ut it was not until 933 that Champernowne [4] gave an explicit example; he showed that the numer N,0 := 0.234567890234567, produced y concatenating the digits of the positive integers, is normal to the ase 0. He also claimed that the numer 0.4689024568202222425, produced y concatenating the digits of the composite integers, is normal to the ase 0, though he did not provide a proof. Such a proof was provided y Copeland and Erdős [5], who showed that if > 2 is an integer and a, a 2, a 3,... is an increasing sequence of positive integers such that for every # < the numer of a n up to N exceeds N # for N su ciently large, then the numer A, := 0.(a ) (a 2 ) (a 3 ) is -normal, where (a n ) denotes the ase- expansion of the integer a n and A := {a, a 2, a 3,...}. In the present day, normality is a property that one presumes a random numer must have. But, of course, just ecause a numer is normal to some ase, does not mean this numer is in any way random. Champernowne s numer N,0 is y no means random. In order to more formally (and quite simply) di erentiate normal from random, Belshaw and Borwein [] introduced a new asymptotic test of pseudorandomness ased on the law of the iterated logarithm, which they call strong normality. Specifically, let 2 (0, ) and m, ( ; n) denote the numer of times that appears in the first n -ary digits of the ase- expansion of. The numer is called simply strongly normal to the ase (or -simply strongly normal) provided oth lim sup n! p m, ( ; n) n/ p = 2n log log n and lim inf n! m, ( ; n) n/ p = 2n log log n p, for every 2 {0,,..., }. This numer is then called strongly normal to the ase (or -strongly normal) if it is m -simply strongly normal for all positive integers m. To illustrate the value of their new asymptotic test of pseudorandomness, Belshaw and Borwein [] showed that a strongly normal (resp. simply strongly normal) numer was also normal (resp. simply normal). They also showed that Champernowne s
INTEGERS: 6 (206) 3 numer N,2, produced y concatenating the positive integers written in inary, is not strongly normal to the ase 2. They commented that this was true for every ase, and a slight modification of their proof indeed gives this result. In this paper, we show that an infinite class of Copeland-Erdős numers is not strongly normal. In fact, our result addresses Bugeaud s Mahler-inspired generalization of the Copeland-Erdős numers. Recall that for a real numer x, xc denotes the greatest integer which is less than or equal to x, and {x} denotes the fractional part of x; that is, x {x} = xc. To ease the exposition, efore stating our result, we introduce a piece of notation in the following definition. Definition. Let c > e a real numer, > 2 e an integer, and a, a 2, a 3,... e an increasing sequence of positive integers. For each n >, let (a n ) denote the ase- expansion of the integer a n. We define the real numer A,,c := 0.(a ) (a ) (a 2 ) (a 2 ) (a 3 ) (a 3 ), where A := {a, a 2, a 3,...} and each loc of -ary digits (a n ) is repeated c`(a n) c times; here `(a n ) is equal to the length of the integer a n written in ase. We prove the following theorem. Theorem. Let c > e a real numer, > 2 e an integer, A N, A(x) e the numer of elements in A that are at most x, and e a real numer satisfying < + log (c ) 2. If for large enough x A(x) 6 then N\A,,c is not -simply strongly normal. x log x, Theorem contains the result of Belshaw and Borwein [] that each Champernowne numer N, = N,, is not -strongly normal. It also yields the following corollary. Corollary. The numer 0.4689024568202222425, produced y concatenating the digits of the composite integers, is not strongly normal to the ase 0. Remar. Bugeaud, in his monograph, Distriution Modulo One and Diophantine Approximation [3, p. 87, Theorem 4.0], was the first to prove the -normality of the generalized Copeland-Erdős numers (with the multiplicities as given in Definition ); the special case of the -normality of N, was proved y Mahler [7].
INTEGERS: 6 (206) 4 2. Non-strong Normality and Generalized Copeland-Erdős Numers In this section, we prove Theorem. Our proof is inspired y Belshaw and Borwein s proof [] of the non--strong normality of the Chapernowne numers N,,. The main idea is to define an increasing sequence (in the index ) of positive integers d,c,, and consider the numer of s in the numer N\A,,c up to the d,c, -th -ary digit. We will show that there is an excess of s. To this end, we start with some properties aout the special sequence over which we will eventually tae limits. Lemma. Let c > e a real numer. For the integers > 2 and >, let d,c, denote the numer of -ary digits in the non-decreasing concatenation of the first 2 positive integers written in the ase, where each integer is repeated c`(a n) c times. Then Moreover, we have d,c, = c c + c n cn n ( ). n= d,c, = c + 2 (c ) (c) + O. Proof. The first assertion of the lemma is immediate. For the second, note that c n c = c n {c n }. Thus! d,c, = c + c n n n ( ) {c } + {c n }n n ( ). Now c( ) n= n= n(c) n = c( ) = c( ) ( ) d X x n dx d dx n=0 x n= x=c x x=c = c( ) (c) (c ) ((c) ) (c ) 2 = c( ) (c) (c) (c) + (c ) 2 c( ) = (c ) 2 (c) c + (c) ( ) = (c ) (c) + O,
INTEGERS: 6 (206) 5 so also, {c } + {c n }n n ( ) = c O( ), n= where c = 0 if c 2 N and otherwise; this accounts for the fact that this term does not appear if c is an integer. Hence d,c, = ( ) (c) + (c ) (c) + O = c + 2 (c ) (c) + O. + c O( ) As our proof will use the comparison of m, ( N\A,,c ; d,c, ) with m, ( N,,c ; d,c, ), we now provide two lemmas. The first provides a way for us to compare these quantities, and the second gives the value of m, ( N,,c ; d,c, ). Lemma 2. If A N and A(x) is as defined in Proposition, then m, ( N\A,,c ; d,c, ) > m, ( N,,c ;d,c, ) c c(a(2 ) A( )) c n cn(a( n ) A( n )). n= Proof. The numer d,c, was defined to e the numer of -ary digits in the concatenation of the first 2 positive integers written in the ase, where each integer is repeated c`(a n) c times. For n =,...,, there are exactly A( n ) A( n ) integers in A of length n and there are a total of A(2 ) A( ) of length. The maximal contriution these numers could have is if each of the numers in A had all of its -ary digits equal to. The inequality in the lemma follows immediately from this oservation. Lemma 3. Let d,c, e as defined in Lemma. Then for large enough m, ( N,,c ; d,c, ) d,c, = (c) 2 + 2 (c ) + co c, where c = 0 if c 2 N and otherwise. Proof. Recall that the numer d,c, denotes the numer of -ary digits in the nondecreasing concatenation of the first 2 positive integers written in the ase, where each integer is repeated c`(a n) c times. Note also, that in the set of numers of length n with any single leading -ary digit, the frequency in the non-leading
INTEGERS: 6 (206) 6 digits of any specific -ary digit is exactly /. Thus m, ( N,,c ; d,c, ) = c n c +c c n= (n ) n ( ) {z } the numer of non-leading s from length n numers ( ) {z } the numer of non-leading s from numers in [, 2 ] = c c + P n= cn cn n ( ) c c + P n= cn c n ( ) Applying the first assertion of Lemma, we have m, ( N,,c ; d,c, ) d,c, which proves the lemma. + n {z} the numer of leading s from length n numers + {z} the numer of leading s! from numers in [, 2 ] + c n c n. n= = c c + P n= cn c n ( ) + = c c + c n c n n= c n c n n= = c c + c (c) n + {c n } n n= n= = c + c (c) + c O( ) c = (c) 2 + (c) c 2 + c O( ) c c = (c) 2 + 2 (c ) + co c, The next lemma provides a su cient (ut not necessary) condition for the non-simple strong normality of the numer N\A,,c. Lemma 4. Let c > e a real numer, A N, d,c, e as defined in Lemma, and suppose that for large enough, we have d,c, m, ( N\A,,c ; d,c, ) > r(c) for some positive constant r. Then N\A,,c is not -simply strongly normal. In particular, N\A,,c is not -strongly normal.!
INTEGERS: 6 (206) 7 Proof. Using the assumptions of the lemma, we have m, ( N\A,,c ; d,c, ) d,c, / r(c) p > p. 2d,c, log log d,c, 2d,c, log log d,c, Now using the second assertion of Lemma, for any " > 0 and large enough, p 2d,c, log log d,c, 6 d,c, 2 /2+" (c ) 6 (c) /2+". c + 2 Thus m, ( N\A,,c ; d,c, ) d,c, / lim p > lim! 2d,c, log log d,c,! r(c) =, ((c) /2+" ) whence the ound on the limit supremum in the definition of -simple strong normality cannot hold for the numer N\A,,c. In addition to the aove results, we will need the following classical result, which is an easy exercise for the curious reader. Lemma 5 (Ael Summation). If a n, n 2 C for all n 2 N, then for all > a n ( n+ n ) = a + + a n+ (a n+ a n ). n= n= We mae use of Ael summation in the following way. Corollary 2. If a n, n > 0 for all n and {a n } n>0 is non-decreasing, then a n ( n+ n ) 6 a + +. n= With all of the aove preliminaries finished, we are now ale to present the proof of Theorem. Proof of Theorem. Applying Lemmas 2 and 3 along with Corollary 2 and the removal of some positive terms, we have m, ( N\A,,c ; d,c, ) > m, ( N,,c ; d,c, ) > d,c, c n n(a( n ) A( n )) n= c (A(2 ) A( )) + (c) 2 + 2 (c ) + co c c A( ) c A(2 ) A( ).
INTEGERS: 6 (206) 8 Thus m, ( N\A,,c ; d,c, ) d,c, > (c) 2 Supposing that A(x) 6 x/ log x, we have m, ( N\A,,c ; d,c, ) d,c, > (c) 2 > (c) 2 + c + c + co c c 2 log + c + co c + log 2 c A(2 ). ( ) log 2 log + O!. Applying Lemma 4 to this last inequality, we have for any suset A N such that < + log = c 2 + log (c ) 2, the numer N\A,,c is not -simply strongly normal. 3. Some Open Questions and Thoughts for the Future In this paper, we showed that a large class of Copeland-Erdős numers is not strongly normal. The main thrust of our argument was the comparison of our class with the Champernowne (or generalized Champernowne) numer. This argument yielded nice results, ut there is a lot left to explore here. Indeed, it would e reasonale to suppose that the decimal formed y the sequence of prime numers is also not strongly normal to the scale of ten, ut of this we have no proof. The following few avenues and questions may lead to some new understanding of oth normality and strong normality as well as their relationship. First, our argument addresses N\A,,c where A N is a su ciently thin set. For example, if A is the set of primes our method wors, ut if A is larger than that in any real asymptotic sense, then our argument fails. In contrast to this, the Copeland-Erdős normality result holds for a much larger class of numers. Question. Do there exist strongly normal Copeland-Erdős numers? Second, Davenport and Erdős [6] proved that if f(x) is a (non-constant) polynomial in x, such that f(n) 2 N for all n 2 N, then the real numer 0.(f()) (f(2)) (f(3)) is normal to the ase. Their result was later generalized in many ways y Naai and Shioawa [8, 9, 0, ]. This egs the following question.
INTEGERS: 6 (206) 9 Question 2. Let f(x) e a (non-constant) polynomial in x, such that f(n) 2 N for all n 2 N. Can the numer 0.(f()) (f(2)) (f(3)) e -strongly normal? Naai and Shioawa [] showed that if f(x) is a (non-constant) polynomial in x, such that f(n) 2 N for all n 2 N, then the real numer (f, ) := 0.(f(2)) (f(3)) (f(5)) (f(p)), where p runs through the prime numers, is -normal. Question 3. Is the numer (f, ) strongly normal for any choice of f and? Finally, the following question, while not specifically aout strong normality, is certainly evident given the existing literature, though it has not explicitly een formulated efore. Question 4. Let > 2 e an integer and a, a 2, a 3,... e an increasing sequence of positive integers such that the numer 0.(a ) (a 2 ) (a 3 ) is -normal. If f(x) is a (non-constant) polynomial in x, such that f(n) 2 N for all n 2 N, then is the numer 0.(f(a )) (f(a 2 )) (f(a 3 )) -normal? References [] Adrian Belshaw and Peter Borwein, Champernowne s numer, strong normality, and the X chromosome, In Computational and analytical mathematics, volume 50 of Springer Proc. Math. Stat., pages 29 44. Springer, New Yor, 203. [2] E. Borel, Les proailités dénomrales et leurs applications arithmétiques, Palermo Rend. 27 (909), 247 27. [3] Yann Bugeaud. Distriution Modulo One and Diophantine Approximation, volume 93 of Camridge Tracts in Mathematics, Camridge University Press, Camridge, 202. [4] D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc. 8 (933), 254 260. [5] Arthur H. Copeland and Paul Erdős, Note on normal numers, Bull. Amer. Math. Soc. 52 (946), 857 860. [6] H. Davenport and P. Erdős, Note on normal decimals, Canadian J. Math. 4 (952), 58 63. [7] Kurt Mahler, Üer die Dezimalruchentwiclung gewisser Irrationalzahlen, Mathematica B, Zutphen 6 (937), 22 36. [8] Yoshinou Naai and Ieata Shioawa, A class of normal numers, Japan. J. Math. (N.S.) 6 (990), no., 7 29. [9] Yoshinou Naai and Ieata Shioawa, A class of normal numers, II, In Numer theory and cryptography (Sydney, 989), volume 54 of London Math. Soc. Lecture Note Ser., pages 204 20, Camridge Univ. Press, Camridge, 990.
INTEGERS: 6 (206) 0 [0] Yoshinou Naai and Ieata Shioawa, Discrepancy estimates for a class of normal numers, Acta Arith. 62 (992), no. 3, 27 284. [] Yoshinou Naai and Ieata Shioawa, Normality of numers generated y the values of polynomials at primes, Acta Arith. 8 (997), no. 4, 345 356.