PROCEEDINGS OF THE
AMERICAN MATHEMATICAL
Volume
91, Number
3, July
SOCIETY
1984
ON MAXIMA OF TAKAGI-VAN DER WAERDEN FUNCTIONS
YOSHIKAZU BABA
ABSTRACT.
Generalizing
Takagi's function ^2(1) and van der Waerden's
function Fio(x), we introduce a class of nowhere differentiable continuous functions FT(x), r > 2. Some properties of Fr(x) concerning especially maxima
are discussed. When r is even, the Hausdorff dimension of the set of i's giving
the maxima of Fr(x) is proved to be 1/2.
1. Introduction.
Let d(x) be the distance from x to the nearest integer. The
function d(x) is continuous and periodic with period 1. Fix an integer r > 2
and define F™(x) — Y^k=o^{rkx)lrkWhen n tends to infinity, F?(x) converges
uniformly to a continuous and periodic (with period 1) function Fr(x). Further,
Fr(x) is proved to be everywhere nondifferentiable.
As a simple example of a
nowhere differentiable continuous function, T. Takagi [1] discovered Fz(x) and a
quarter of a century later B. L. van der Waerden [2] rediscovered i*\n(x). Takagi's
proof of the nowhere differentiability of F2 (x) is applicable to any r > 3 with a slight
modification when r is odd. Recently, B. Martynov [3] discussed the structure of
the set E2 = {0 < x < l;F2(x) = M2} where Mi = maxF2(x) and the result is
that x = O.X1X2• • • xn ■■■ (the base-4 expansion of x) belongs to Ei if and only if
xn = 1 or 2 for any n > 1. From this result we can easily see that the Hausdorff
dimension of E2 is equal to log 2/log 4 = 1/2. This has a relation to the fact that
the Hausdorff dimension of the set of zeros of the Brownian motion B(t, w) is equal
to 1/2 and the sample functions of B(t,w) are nowhere differentiable continuous
ones for almost all w. In this paper we show that for any even r > 2 the Hausdorff
dimension of the set Er = {0 < x < l;Fr(a;) = Mr = maxiV(x)}
is equal to 1/2
generalizing Martynov's arguments to r > 2.
2. Functional
equations.
PROPOSITION 1. The function Fr(x) satisfies the following functional equations:
(1)
Fr(rx) =rFr(x)
(2)
Fr{x) = F}(x) + ^FT{r2x).
Proof.
First,
Fr"(rx)
_
-rd(x),
= r{d{rx)/r
+ ■■■+ d{rn+1x)/rn+1)
=r(i7+1(x)-d(*)).
Received by the editors September 9, 1983.
1980 Mathematics Subject Classification. Primary 26A27; Secondary 60G17.
Key words and phrases. Nowhere differentiable
continuous
function,
©1984
Hausdorff
American
0002-9939/84
373
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dimension.
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YOSHIKAZU BABA
374
Taking the limit of the both sides, we have (1). Next,
Fr(r2x)
=* rFr(rx)
- rd(rx)
- r(rFr{x)
- rd(x)) - rd{rx)
= r2(Fr(x)-F}{x)).
This implies (2).
PROPOSITION 2. The function Fr{x) is the unique bounded solution of the functional equation
(1')
f{rx) = rf{x) - rd{x).
PROOF. Substituting rk~1x for x in (1') and dividing both sides of the resulting
equation by rk, we have
f{rkx) _ f{rk~l:
dir^x)
rfe-l
Summing up these for k — 1 to n, we have
ra-l
f(rnx)
m-E
d{rkx)
k=0
Letting n —>co in the both sides, we obtain f(x) = Fr(x).
REMARK. The functional equation (1') for r = 2 is a special case of the functional equation studied by M. Yamaguti and M. Hata [4].
3. Er and Mr. Observing the graphs of the functions
can easily see that (i) if r is odd, then Er = {1/2} and
1A 1 1
= = (! + - + -* +
Mr=Fr
F}{x),
F2(x),...,
we
2r-2
and (ii) if r is even, then F*(x) — 1/2 for all x in the interval [(r- l)/2r, (r + l)/2r].
Therefore, in the following, we consider only the case of even r's.
Proposition
(3)
3.
Er C
r
r+ 2
2r + 2'2r + 2
c
r- 1 r + 1
2r
2r
,2
(4)
Mr =
2r2-2
V ~T\2r + 2j
PROOF. First, by the periodicity
= Fr
'"^
r+ 2
+ 2,
of the function Fr(x), we have
1
2r + 2
Fr
2r + 2
uu
2r + 2
2r + 2
Therefore Fr{r/{2r + 2)) = r2/(2r2 - 2) and this is equal to Fr{{r + 2)/(2r + 2))
since the function Fr(x) is symmetric with respect to 1/2. Next, take x G Er with
x < 1/2. Then Fr{rx) = rFr(x) - rd(x) = rFr(x) - rx < Mr, therefore we have
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ON MAXIMA OF TAKAGI-VAN DER WAERDEN FUNCTIONS
From this and by the symmetric
x G Er, then
property
375
of Fr(x) follows (3). To obtain (4) take
Mr = Fr{z) = F*(s) + ^Fr(r2x)
<± + ±Mr.
From this follows
i
r2
2(1-1/r2)
/
2r2-2
-
■\
\2r + 2,
obtaining (4).
PROPOSITION 4. (i) If x G Er, then the fractional part of r2x (= {r2x}) also
belongs to Er.
(ii) IfFr{r2x) = Mr and ie[(rl)/2r, (r + l)/2r], then x G £r.
Proof,
(i) If x g Er, then
2(^1)
Therefore Fr(r2x)
= r2/2(r2
ofFr(x).
(ii) By the assumptions
Fr(x)=1-
= *M = \ + ^r(r2a:).
- 1) from which follows {r2x} G Er by the periodicity
we have
+ ±Fr(r2x)=1-
2
+
2(r2-l)
2(r2-l)'
obtaining x 6 Er.
PROPOSITION 5. Suppose x G Er and k is a positive integer.
{x + k)/r2 G Er if and only if (r2 - r)/2 < k < (r2 +r- 2)/2.
Then we have
PROOF. If (x + k)/r2 e Er, then
r
x + rcr
2r + 2~
r2
+ 2
-2r
+ 2'
To satisfy these inequalities, it is necessary for k to be in the interval [(r2 —r)/2,
{r2+r2)/2] as is easily checked. Conversely, if (r2 - r)/2 < k < (r2 + r - 2)/2,
then for x G Er the inequalities
r —1
x + fc
r+1
2r - ~r2~ - 2r
hold as is easily seen and Fr(r2(x + k)/r2) = Fr{x + k) = Mr because of the
periodicity of Fr(x). By (ii) of Proposition 4 follows (x + k)/r2 G Er.
REMARK. The number of the integers k satisfying the inequalities (r2 —r)/2 <
fc < (r2 + r - 2)/2 is equal to r.
4. Ba8e-r2 expansion.
Put a = (r2 - r)/2 and /9 = (r2 + r - 2)/2. Let us
expand the smallest and the largest numbers of Er into base-r2 decimals (r2-mals).
Then
r
2r + 2
= I^
/ ox IQ = 0Qa• • • a• • ■= O.à
and
r+ 2
2r + 2
-g^)í-<W-/.--a#.
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376
YOSHIKAZU BABA
Generally we have the following proposition.
PROPOSITION 6. Let x = 0.xix2 • • • xn ■■■be the base-r2 expansion of x G [0,1].
Then, x G Er if and only if a < xn < ß for any n > 1.
PROOF. Sincer/(2r + 2) and (r + 2)/(2r + 2) belong to Er, if integers xi,x2,...,
xn are contained in the interval [a, ß], then by applying n times (ii) of Proposition 4,
we have 0.xix2 • • • xnà G Er and 0.xix2 • ■• xnß G Er. From this follows 0.xix2 • • •
xn • • • G Er by the continuity of Fr{x). Conversely, if there exists an integer xn £
[a, ß] in the base-r2 expansion O.XiX2 • • • xn ■■■G Er, then by using n - 1 times (i)
of Proposition 4, we have 0.xnxn+\ ■■■G Er and this contradicts to Er C [O.à, 0.$}.
5. Hausdorff
THEOREM.
dimension.
For any even integer r > 2, we have dimEr
= 1/2.
PROOF. For all positive integers n, Er can be covered by rn, and no fewer than
rn, intervals of length r~2n. Therefore, for every positive real number s < 1, Er
can be covered by rn+1, and no fewer than rn, intervals of length s, where n —
[-logs/2logr].
But
lim log?•"/(-logs)
s—»0
= lim logrn+l/(-logs)
s—0
= 1/2,
so dim Er = 1/2.
ACKNOWLEDGEMENT.I am grateful to the referee for simplifying the proof of
the theorem.
REFERENCES
1. T. Takagi, A simple example of the continuous function without derivative, Proc. Phys.-Math.
Soc.
Tokyo Ser. II 1 (1903), 176-177.
2. B. L. van der Waerden,
Fan einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion, Math.
Z. 32 (1930), 474-475.
3. B. Martynov,
4. M. Yamaguti
On maxima of the van der Waerden function, Kvant, June 1982, 8-14. (Russian)
and M. Hata, Weierstrass's junction and chaos, Hokkaido Math. J. (to appear).
FACULTY OF LIBERAL ARTS, SHIZUOKA UNIVERSITY, SHIZUOKA 422, JAPAN
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