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On the digits in the base‐ b expansion ofsmooth numbers
Yann Bugeaud
Institut de Recherche Mathčmatique Avancče, U.hf.R. 7501
Université de Strasbourg et C.N.R.S.
Hajime Kaneko
Institute of Mathematics, University of Tsukuba
Research Core for Mathematical Sciences, University of Tsukuba
Abstract\acute{}Erdó s [4] conjectured that, for any integer m\geq 9 , the digit 2 appears at least oncein the ternary expansion of 2^{m} . More precisely, Dupuy and Weirich [3] conjectured
that. for any sufficiently largem, the digits 0,ı, and 2 appear “uniformly” in theternary expansion of 2^{m} . This is still open. Stewart [10] obtained a lower boundfor the number of nonzero digits in the ternary expansion of 2^{m} , thus giving (very)
partial results of ‘uniformity In this report, we investigate the number of nonzero
digits in the base‐b expansion of more general smooth numbers and introduce themain results established in [2].

1Problems on the base‐b expansion of a^{n}

Throughout this survey, b denotes an integer greater than 1. The main purpose of thisreport is the study of the uniformity of the digits in the base‐b expansion of smooth

numbers. We now recall the definition of smooth numbers. We denote by P[n] thegreatest prime factor of an integer n\geq 2 . For convenience, let P[1] :=1 . Let x bea positive real number. Recall that a positive integer n is x ‐smooth if P[n]\leq x . For
instance, let a be an integer greater than 1. Then, a^{m} is P[a] ‐smooth for any nonnegative
integer m . In this section we review open problems related to the base‐b expansions ofpowers of integers.
\acute{}In 1979, Erdó s [4] conjectured that, if m is an integer greater than 8, then the digit 2appears at least once in the ternary expansion of 2^{m} (note that 2^{8}=3^{\overline{D}}+3^{2}+3+1 ). Let
T be a positive integer. Denote by N(T) the number of integers m with 0\leq m\leq T suchthat the ternary expansion of 2^{m} omits 2. It is still unproven whether N(T) is bounded
as T tends to infinity. In 1980, Narkiewicz [7] showed thatN(T)\leq 1.62T^{\log_{3}2},

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where \log_{3}2= (log2)/(1og3) \approx 0.63092 . Moreover, for any positive real number \lambda_{:} let
N_{\lambda}(T) be the number of integers m with 0\leq m\leq T such that the ternary expansion of\lflo r\lambda 2^{m}\rflo r omits 2. Lagarias [6] proved that

N_{\lambda}(T)\leq 25T^{0.9725}.
for any T\geq T_{0}(\lambda) , where T_{0}(\lambda) is a positive number depending only on \lambda.
Recall that two positive integers a and b are multiplicatively independent if a^{i}b^{j}\neq 1 forany integers (i, j)\neq(0,0) . Let a and b be multiplicatively independent positive integers.
Let v=v_{1}v_{2}\ldots v_{\iota} be any finite word over the alphabet \{0,1, . . . , b-1\} of length l\geq ı.

Lagarias [6] conjectured that v occurs at least once in the base‐b expansion of a^{m} forany sufficientlč large integer m . Erdós conjecture treats the casel=1 .

We introduce arelated result from [5]. Let p be a prime number and a\geq 2 an integer coprime to p . Let
v=v_{1}v_{2}\ldots v_{i} be any finite word over the alphabet \{0.1_{:L} .p-1\} with length l\geq 1.
Let \gamma(\geq 2) be the number of circular shift occurrences of v in vv=v_{1}v_{2}\ldots v_{i}v_{1}v_{2}\ldots v_{l}.
Moreover, let e_{p}(v:_{!}a^{m}) be the number of (possibly overlapping) occurrences of v in thebase‐p expansion of a^{m} . It is shown in [5] that we have

1 i_{M}\sup_{ar ow\infty}\frac{p(v;a^{m}) {\log m}\geq\frac{\gamma-1}{l\log p}.
It efining E\cdot d". conj (^{\backslash },(^{\backslash }.tn1^{\cdot}(), Dn ]) ny_{\dot{c}11( }dW\backslash irich [3] ]) 1^{\cdot}()1 ) O_{\iota}*(^{\backslash}. ( th_{(}\backslash. following 1 )robl
on th (\backslash.
uniformity of digits. Let p and q be distinct prime numbers. Let h be an integer with
0\leq h\leq p-1 . Then Dupuy and Weirich [3] conjectured that\backslash\lim_{mar ow\infty}\frac{e_{p}(h;q^{m}) {\log_{p}(q^{m}) =\frac{ \imath} {p}.
Moreover, they obtained the following partial results in the direction of the conjecturestated above. Let kI ) e a fixed 1 )( s^{\backslash} itive integer. \Gamma_{o1\dot{c} ) .ny integer m, h with 0\leq m, 0\leq h\leq
p-1 , letq^{m}=s_{0}+\mathcal{S}_{1}(p)(p)_{p+\cdots+s_{M}^{(p)}p^{\Lambda\prime}}be the base‐p expansion of q^{m} , where0,Then

M=\lfloor\log_{p}(q^{m})\rfloor and s_{i}^{(p)}\in\{0,1, , p-1\} for i=, M . Let e_{p}(h, k;q^{m}) be the number of occurrences of h in the word s_{k-1}^{(p)}\ldots s_{1}^{(p)}s_{0}^{(p)}.

\lim_{kar ow\infty}\lim_{Nar ow\infty}\frac{1}{N}\sum_{m=1}^{N}\frac{e_{p}(h, k;q^{m}) {k}=\frac{1}{p}.

It is in general very difficuıt to get some non‐trivial information on the number of oc‐
currences e_{p}(h;q^{m}) of a fixed digit h in q^{m} . In the next section we consider the numberof nonzero digits, which also gives partial results for the uniformity of digits. In Section
2, we introduce various results on the number of nonzero digits of smooth numbers. InSection 3, we present the main new results obtained in [2], which improve and extend theresults of Section 2.

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2

Number of nonzero digits of smooth numbers

For any positive integer n , let \lambda_{b}(n) be the number of nonzero digits in the base‐b expansionof n . If the conjecture by Dupuy and Weirich in Section 1 is true, thenmar ow\infty1\dot{ \imath} m\frac{\lambda_{p}(q^{m}) {\log_{p}(q^{m}) = \frac{p-1}{p}.
However, it is still unknown whether

1 i_{M}\sup_{ar ow\infty}\frac{\lambda_{p}(q^{m}) {\log_{p}(q^{m}) >0.
We introduce the lower bounds for the number of nonzero digits established by Stewart[10]. Let a and b be multiplicatively independent integers greater than 1. Let \varepsilon be an
arbitrary positive real nu1nber. Then Stewart [10] showed that there exists an effectively
computable positive number C_{1}(a_{:}b_{\dot{\ovalbox{\t \smal REJECT} \varepsilon) such that\lambda_{a}(n)+\lambda_{b}(n)\geq(1-\varepsilon)\frac{\log\log n}{\log\log\log n},
for any n\geq C_{1}(a, b, \varepsilon) . In particular, consider the case of n=a^{m} , where m is a nonnega‐tive integer. Since \lambda_{a}(a^{m})=1 , we see that\lambda_{b}(a^{m})\geq(1-\varepsilon)\frac{\log\log a^{m} {\log\log\log a^{m} ,(2.1)

that is,\lambda_{b}(a^{m})\geq(1-\varepsilon)\frac{\logm}{\log{\imath}ogm} \dot{\ovalbox{\t \smal REJECT}
for any sufficiently large integer m . The proof rests on a subtle use of estimates forcomplex linear forms in the logarithms of rational numbers.
Lower bounds for the number of nonzero digits in the base‐b expansion of more generalsmooth numbers were discussed in [1]. Note that \lambda_{b}(bn)=\lambda_{b}(n) for any positive integer. Thus, we assume that n is not divisible by b in the rest of this section. It is easy to seethat if n\geq b+1 , then \lambda_{b}(n)\geq 2 , which is a triviaı lower bound. Thus, we first consider
sufficient conditions for \lambda_{b}(n)\geq 3 and \lambda_{b}(n)\geq 4 . Note that \lambda_{b}(n)=2 if and only if n has
the form n=t_{1}b^{m}+t_{0} , where m is a positive integer and t_{1} , t_{0}\in\{1,2, . . . , b-1\} . Applyingn

the result on the greatest prime factor of linear recurrences established by Stewart [9], weobtain that there exists an effectively computable positive number C_{2}(b) , depending only
on b , such that if an integer n\geq C_{2}(b) satisfies

P[n] \leq(\log n)^{1/2}\exp(\frac{\log\log n}{10S\log 1\cdot g\log n}),then \lambda_{b}(n)\geq 3 . In the opposite direction, Schinzel [8] constructed arbitrary large integersnsuch that

P[n]\leq n^{c/\log\log\log n} and \lambda_{b}(n)=2,

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wherecis an absolute real number.

Furthermore, the following resuıt was obtained in [1]. Let \varepsilon be an arbitrary positive real
number. Then there exists an effectively computable positive number C_{3}(b, \varepsilon) , depending
only on b, \varepsilon , such that if n\geq C_{3}(b, \varepsilon) satisfies
P[n]\leq(1-\varepsilon) (log log n )\frac{\log{\imath} og\log n}{\log\log\log\log n} ,(2.2)

then \lambda_{b}(n)\geq 4 . In addition, it was also proved in [1], that, for any fixed integer N\geq 4,
the greatest prime factor of n tends to infinity, as n tends to infinity and runs throughthe set of integers not divisible by b and having at most N nonzero digits in their base‐bexpansion. Since the proof rests on the subspace theorem, no estimate for the speed ofdivergence can be derived, thus no sufficient condition on P[n] ensuring that \lambda_{b}(n)\geq 5 wasknown until very recently. In the next section we give sufficient condition for \lambda_{b}(n)\geq k_{:}where k is an arbitrary positive integer.

3Main results

In this section, we review without proof lower bounds for \lambda_{b}(n) obtained in [2]. Through‐out this section, C_{i}(x, y\ldots)(i=4,5, \ldots) denote effectively coln putable positive numbers
depending only on x, y , .

THEOREM 3.1. Let k be an integer greater than 2. Let \varepsilon be an arbitrary positive real
number. Let n\geq C_{4}(b, k, \varepsilon) be an integer not divisible by b . Suppose that

P[n] \leq(\frac{1}{k-2}-\varepsilon)(\log\log n)\frac{\log\log\log n} {\log\log\log\log n} .
Then we have

(3.1)

\lambda_{b}(n)\geq k+1.

Note that (3.1) generalizes (2.2). Now let \mathcal{A} be the set of positive integers n notdivisible by b such that n is \log\log n‐smooth. Applying Theorem 3.1, we see\lim_{n\in A,narrow\infty} \lambda_{b}(n)=\infty.
In what follows, we investigate quantitative version of this result, that is, we give lower
bounds\lambda_{b}(n)\geq\varphi(n)(3.2)

for smooth numbers n , using suitable increasing functions \varphi with \lim_{narrow\infty}\varphi(n)=\infty.
THEOREM 3.2. For any n\in \mathcal{A} with n\geq C_{5}(b) , we have\lambda_{b}(n)\geq\frac{1}{2}\cdot\frac{\log\log\log n}{\log\log\log\log n}.
Changing the condition of smoothness, we give another lower bound of the form (3.2).

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THEOREM 3.3. Let n\geq C_{6}(b) be an integer not divisible by b . Suppose that

P[n] \leq(\log\log n)^{1/2}(\frac{\log\log\log n}{\log\log\log\log n})^{1/2}Then, we have\lambda_{b}(n)\geq\frac{1}{3}(\log\log n)^{ \imath}/2}(\frac{\log\log\log n} {\log\log\log\log n})^{1/2}
We now generalize (2.1). Let S be a non‐empty finite set of prime numbers. Recall
that a positive integer n is an integral S‐unit if all the prime factors of n are in S . In
particular, if p denotes the maximal element of S , then any integral S ‐unit is p‐smooth.

THEOREM 3.4. Let S be a non‐empty finite set of prime numbers. Let \varepsilon be an arbitrary
positive real number. Let n\geq C_{7}(b_{:}S_{:}\varepsilon) be an integral S ‐unit not divisible by b . Then wehave\lambda_{b}(n)\geq(1-\varepsilon)\frac{\log\log n}{\log\log\log n} .
Let a\geq 2 be an integer coprime to b . Let

S be the set of prime divisors of a .

(3.3)
Then,(2.1) follows from TheoreIn 3.4.
Changing the coefficient of the right‐hand side of (3.3). we can extend (2.1) to moregeneral smooth numbers.
THEOREM 3.5. Let n\geq C_{8}(b) be an integer not divisible by b . Suppose that

P[n] \leq\frac{1}{2}(\log\log\log n)\frac{\log\log\log\log n} {\log\log\log\log\log n}.
Then, we have\lambda_{b}(n)\geq\frac{1}{2}\frac{\log\log n}{\log\log\log n} .

(3.4)

Note that (3.4) is the best lower bound (upto the value \frac{1}2 ) for \lambda_{b}(n) obtainable by ourmethod. We conclude with a general statement giving lower bounds for \lambda_{b}(n) .

THEOREM 3.6. Let f be a positive real valued function defined over the set of positive
integers. Assume that

\lim_{narrow\infty}f(n)=\infty
and that there exists a real number 0<\delta<1 satisfying

f(n)\leq (ı— \delta )\frac{\log\log n}{\log\log\log n}

for any sufficiently large integer n . Set\Psi_{f}(n):=\frac{\log\log n}{f(n)}

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and

\delta_{0}^{-}:= \sup\{\delta>0 : f(n) \leq(1-\delta)\frac{\log\log n}{\log\log\log n} for(1?r(/s(Lffi_{C.(}inll\cdot(/lar.(7^{(}n}.

Let \varepsilon be an arbitrary positive real number. Suppose that a sufficiently large integer n notdivisible by b satisfies

P[n] \leq(\delta_{0}^{-} \varepsilon)\Psi_{f}(n)\frac{\log\Psi_{f}(n)}{\log\log \Psi_{f}(n)}.
Then we have\lambda_{b}(n)\geq f(n).

The proofs of all the results stated in this section depend on lower estimates for linear
forms in the complex logarithms of rational numbers, combined with lower estimates forlinear forms in the p‐‐adic logarithms of rational numbers, where p is a prime divisor of b.

Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 15K17505.

References[1] Y. Bugeaud, On the digital representation of integers with bounded prime factors,
preprint arXiv:1609.07926, to appear in Osaka J. Math.

[2] Y. Bugeaud and H. Kaneko, On the digital representation of smooth numbers,
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[3] T. Dupuy and D. E. Weirich, Bits of 3“ in binary, Wieferich primes and a conjectureof Erdó s, J. Number Theor. 158 (2016), 268‐280.
\acute{}[4] P. Erdós, Some unconventional problems in number theory, Math. Mag. 52 (1979),67‐70.

[5] H. Kaneko and T. Stoll, On subwords in the base‐q expansion of polynomial andexponential functions, Integers 18A (2018), Article All.

[6] J. Lagarias, Ternary expansions of powers of 2, J. Lond. Math. Soc. (2) 79 (2009),
no. 3, 562‐588.

[7] W. Narkiewicz, A note on a paper of H. Gupta concerning powers of 2 and 3, Univ.
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