Walking on real numbers
David H Bailey, Lawrence Berkeley National Lab, USA
Collaborators: Francisco J. Aragon Artacho (Univ. of Newcastle,Australia), Jonathan M. Borwein (Univ. of Newcastle, Australia) andPeter B. Borwein (Simon Fraser Univ., Canada)This talk is available at:
http://www.davidhbailey.com/dhbtalks/dhb-walking.pdf

1Normal numbers
Given an integer b > 1, a real number x is b-normal (or “normal base b”) if
every m-long string of digits in the base-b expansion of x appears with
precisely the expected limiting frequency b-m.
Using measure theory, it is can be shown that almost all real numbers are bnormal for a given integer base b. In fact, almost all reals are b-normal
for all integer bases b > 1 simultaneously (i.e., are “absolutely normal”).
These are widely believed to be b-normal, for all integer bases b > 1:
π = 3.1415926535…
e = 2.7182818284…
sqrt(2) = 1.4142135623…
log(2) = 0.6931471805…
Every irrational algebraic number (this conjecture is due to Borel).

But there are no normality proofs for any of these constants in any base, nor
are there any nonnormality results for any of these constants.
Until recently, normality proofs were known only for a few relatively contrived
examples such as Champernowne’s constant =0.123456789101112131415… (which is 10-normal)
2

A result for algebraic numbers
If x is algebraic of degree d > 1, then its binary expansion through position nmust have at least C n1/d 1-bits, for all sufficiently large n and some C that
depends on x.
Example: The first n binary digits of sqrt(2) must have at least sqrt(n) ones.
However, note that these results are still a far cry from full normality, even in
the single-digit sense.

DHB, J. M. Borwein, R. E. Crandall and C. Pomerance, “On the Binary Expansions of Algebraic Numbers,”
Journal of Number Theory Bordeaux, vol. 16 (2004), pg. 487-518.

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New computer-based approaches tothe normality problem
In a recent paper to appear in the Mathematical Intelligencer (see below), weapplied the following to this problem:· Analyses of Stoneham numbers (with proofs or normality and nonnormality), the Erdos-Borwein constants and other classes.
· High-resolution computer graphics.
· Representation of digits as a “random” walk.
· Statistical studies based on the random walk data.
· Fractal dimension analysis.
· “Strong” normality versus ordinary normality.
· Applying these techniques to analyses of genome sequences.

Francisco J. Aragon Artacho, David H. Bailey, Jonathan M. Borwein and Peter B. Borwein, “Walking on realnumbers,” Mathematical Intelligencer, to appear (Jan 2013), available at
http://www.davidhbailey.com/dhbpapers/tools-walk.pdf
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A class of provably normal constants
DHB and Richard Crandall have shown that an infinite class of constants is bnormal (and thus bm-normal for any positive integer m), for instance:
2,3

=

1
n 23n
3
n=1

=0.041883680831502985071252898624571682426096 . . .10=

0.0ab8e38f 684bda12f 684bf 35ba781948b0f cd6e9e0 . . .16

This particular constant was proven 2-normal by Stoneham in 1971. We
extended this to the case where (2,3) are any pair (b,c) of relatively primeintegers > 1, and also to an uncountable class (here rn is n-th bit of r in [0,1)):1(r)=2,3n 23n +rn
3n=1
More recently, DHB and Michal Misiurewich established the α2,3 result more
simply by means of a “hot spot” lemma proved using ergodic theory.
DHB and M. Misiurewicz, “A Strong Hot Spot Theorem," Proceedings of the American Mathematical Society,vol. 134 (2006), no. 9, pg. 2495-2501.
DHB and R. E. Crandall, “Random Generators and Normal Numbers,” Experimental Mathematics, vol. 11, no.
4 (2002), pg. 527-546.
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A nonnormality result
Although α2,3 is provably 2-normal, surprisingly enough it is NOT 6-normal.
Note that we can0write1
blog3 nc
1XXmn m n 3m
nn m n 3 A@
mod 1 +32
6 ↵2,3 mod 1 =32
m=1

m=blog3 nc+1

The first portion of this expression is zero, since all of the terms in thesummation are integers. When n = 3m, the second portion is accurately
approximated by the first term of the series. Thus,m63m↵2,3 mod 1 ⇡3 343m+1

Because this is so small for large m, this means the base-6 expansion of
α2,3 has long stretches of zeroes beginning at positions 3m + 1. Thisobservation can be fashioned into a rigorous proof of nonnormality.
DHB and J. M. Borwein, “Normal Numbers and Pseudorandom Generators,” to appear in Heinz Bauschke,ed., Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan
Borwein's 60th Birthday, Springer, 2011, http://crd.lbl.gov/~dhbailey/dhbpapers/normal-pseudo.pdf.
6α2,3 in base-6 digits
0.
013014043000333425113050213000000124355504543223301150024352532055135234354101
043000000000000000051411300540405554553031442504334351012413452351125142125134
505503545015053522052044340452151505102411552500425130051124454001044131150032
420303213000000000000000000000000000000000000000000142120343111214520135254453
421134122402205253010542044235524110554150155204350414555400310145303033532002
534340401301240104453254343502142020432415025555101004043300045544114501031331
451151014451412344334234124005513133350454235305531511535015334524354502500555
214530542343421530350125024205404135451231323245353031534552304115020154242121
145201542222534340340450530123325534440443103332445332141415014233454542412432
031253400501341502455144043000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000031335054
244443111105553414105201454021341231300142433313311551144002552443344143003224335300102353515310402031320020533130021221200323413302222122521000035254543002
545040035245112205252324300200332523104022522341033520112253412034105440205323
451011020023525150015212545502421545343134553235001545540400023411521342323335
352445254035541452214132514501442052123331155522222331511535031203434310450421
303512422055300241415240113013242241234152033301205305300410045411552323315404124335430412302302244135255011522313421543013205100505502430344422115032003312043000355240351155014141310244412135431134045402354431451021050554450430513450
125125412054431444230051425001123154412512211541355432123414214030443515204024
235524524554300502531314011015041242313530200005202433402032025411301324022320
213143235351510550435543534053314401324313010031034432330333020541411524554102024511252554525045245533541442124051204242044021002315121441215553532044450502551452010241043242055023411434513124341231325032130205521143454221303254322525
001003215023424404353350251050434103131435231504421350241010405301043142100315114422021202242031254124153534242220443313314424101440231132320424442150031332450441322425500055412431400314402120525313133331045531400355154135352044001245
423040225112500244121430424020245242243022325455215430225105432551034135040520
213000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000005304302452321100553442520302

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General nonnormality results for theStoneham alpha constants
Given co-prime integers b > 1 and c > 1, and integers p, q, r > 0, with neitherb nor c dividing r, let B = bp cq r, and assume this condition holds:

D = cq/p r1/p /bc 1 < 1Then the constant1X

1↵b,c =nncc b
n=1
is not B-normal. Thus, for example, αb,c is b-normal but not bc-normal.
If αb,c and αd,e are two Stoneham constants both nonnormal to base B, asgiven in the previous result, with c and e multiplicatively independent, thenthe sum is also nonnormal base B."
Ref: DHB and J. M. Borwein, “Nonnormality of the Stoneham constants, Ramanujan Journal, to appear,
available at http://crd.lbl.gov/~dhbailey/dhbpapers/nonnormality.pdf.
8Walk on the first 100 billion base-4 digits of pi(0: up; 1: right; 2: down; 3: left)9α2,3 as a “random” walk:base 2 (normal) vs base 6 (nonnormal)

10Champernowne’s number C4(normal base 4)

11

Horizontal color maps of the digits of pi
versus pseudorandom digits

12Color map of α23 base 6 (nonnormal)

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For additional detailsThis talk is available at
http://www.davidhbailey.com/dhbtalks/dhb-walking.pdf
For full details, see:
1. Francisco J. Aragon Artacho, David H. Bailey, Jonathan M. Borwein andPeter B. Borwein, “Walking on real numbers,” Mathematical Intelligencer,
to appear (Jan 2013), available at
http://www.davidhbailey.com/dhbpapers/tools-walk.pdf
2. David H. Bailey and Jonathan M. Borwein, “Nonnormality of Stoneham
constants,” Ramanujan Journal, to appear, available at
http://www.davidhbailey.com/dhbpapers/nonnormality.pdf
3. David H. Bailey, Jonathan M. Borwein, Cristian S. Calude, Michael J.
Dinneen, Monica Dumitrescu and Alex Yee, “An empirical approach to
the normality of pi,” Experimental Mathematics, vol. 21 (2012), pg.
375-384, available at
http://www.davidhbailey.com/dhbpapers/normality.pdf
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