Mathematische Annalen

, Volume 364, Issue 1–2, pp 125–150

# On simply normal numbers to different bases

• Verónica Becher
• Yann Bugeaud
• Theodore A. Slaman
Article

## Abstract

Let $$s$$ be an integer greater than or equal to $$2$$. A real number is simply normal to base $$s$$ if in its base-$$s$$ expansion every digit $$0, 1, \ldots , s-1$$ occurs with the same frequency $$1/s$$. Let $${\mathcal{S}}$$ be the set of positive integers that are not perfect powers, hence $$\mathcal{S}$$ is the set $$\{2,3, 5,6,7,10,11,\ldots \}$$. Let $$M$$ be a function from $$\mathcal{S}$$ to sets of positive integers such that, for each $$s$$ in $$\mathcal{S}$$, if $$m$$ is in $$M(s)$$ then each divisor of $$m$$ is in $$M(s)$$ and if $$M(s)$$ is infinite then it is equal to the set of all positive integers. These conditions on $$M$$ are necessary for there to be a real number which is simply normal to exactly the bases $$s^m$$ such that $$s$$ is in $$\mathcal{S}$$ and $$m$$ is in $$M(s)$$. We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases.

### References

1. 1.
Becher, V., Slaman, T.A.: On the normality of numbers to different bases. J. Lond. Math. Soc. 90(2), 472–494 (2014)
2. 2.
Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Supplemento Rendiconti del Circolo Matematico di Palermo 27, 247–271 (1909)
3. 3.
Bugeaud, Y.: Distribution Modulo One and Diophantine Approximation. Cambridge Tracts in Mathematics, vol. 193. Cambridge University Press, Cambridge (2012)Google Scholar
4. 4.
Bugeaud, Y.: On the expansions of a real number to several integer bases. Revista Matemática Iberoamericana 28(4), 931–946 (2012)
5. 5.
Cassels, J.W.S.: On a problem of Steinhaus about normal numbers. Colloq. Math. 7, 95–101 (1959)
6. 6.
Eggleston, H.G.: Sets of fractional dimensions which occur in some problems of number theory. Proc. Lond. Math. Soc. Second Ser. 54, 42–93 (1952)Google Scholar
7. 7.
Haiman, M.: Private correspondence (2013)Google Scholar
8. 8.
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008)Google Scholar
9. 9.
Hertling, P.: Simply normal numbers to different bases. J. Univ. Comput. Sci. 8(2), 235–242 (2002, electronic)Google Scholar
10. 10.
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover (2006)Google Scholar
11. 11.
Long, C.T.: Note on normal numbers. Pac. J. Math. 7, 1163–1165 (1957)
12. 12.
Maxfield, J.E.: Normal $$k$$-tuples. Pac. J. Math. 3, 189–196 (1953)
13. 13.
Pollington, A.D.: The Hausdorff dimension of a set of normal numbers. Pac. J. Math. 95(1), 193–204 (1981)
14. 14.
Schmidt, W.M.: On normal numbers. Pac. J. Math. 10, 661–672 (1960)
15. 15.
Wolfgang, M.: Schmidt. Über die Normalität von Zahlen zu verschiedenen Basen. Acta Arith. 7, 299–309 (1961/1962)Google Scholar

## Authors and Affiliations

• Verónica Becher
• 1
• Yann Bugeaud
• 2
• Theodore A. Slaman
• 3
1. 1.Universidad de Buenos AiresBuenos AiresArgentina
2. 2.Université de StrasbourgStrasbourgFrance
3. 3.University of California BerkeleyBerkeleyUSA