On badly approximable functions

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JOURNAL OF APPROXIMATION THEORY 17, 28&296 (1976)
On Badly Approximable Functions*
T. W. GAMELIN
Department of Mathematics, University of California, Los Angeles
J. B. GARNETT
Department of Mathematics, University of California, Los Angeles
L. A. RUBEL
Department of Mathematics, University of Illinois at Urbana-Champaign, Illillois
AND
A. L. SHIELDS
Department of Mathematics, University of Michigan, Ann Arbor
Commkcated by G. G. Lorentz
Received November 2, 1974
1. INTRODUCTION
Let D be a bounded domain in the complex plane with boundary r, and let A(D) be the algebra of analytic functions on D which extend continuously to r. The distance from a function v E C(r) to A(D) is defined to be
4~ A(D)) = W F -fll :f~ A(D)),
where the norm is the supremum norm over r. In this paper, we consider the problem of describing the functions 9) E C(I’) which satisfy
Such functions, excepting the function 0, will be called badly approximable. Thus a function is badly approximable if its best analytic approximant is 0.
* This work was supported by grants from the National Science Foundation. 280
Copyright Q 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.

ON BADLY APPROXIMABLE F-UNCTIONS 23s
For q~ a nonvanishing function on r, there is a unique integer m with the following property: there is a continuous nonvanishing function f on ~3 such that for z,, E D, the function sjfi(z - z~)~% has a continuous logarithm on r. The integer m is called the index of q~, and denoted by ind(y). If I consists of a finite number of simple closed disjoint Jordan curves, then ind(F) is the usual winding number of v around I-Y
Qur aim in this paper is to prove simply and to extend to more genera’. domains, the following theorem of Poreda [lo].
P0m~A.s THEOREM. Suppose r consists of u simple c,‘osed Jordan curve. Then 9 E G(T) is badly approximable if and only <f y has nonzero corzstant modulus and ind(F) < 0.
Half of Poreda’s theorem extends trivially to arbitrary domains, as fohows.
THEOREM 1. I. lfy E C(r) has nonzero constant modulu, a~d $ind(F) < 0, then 50 is badly approximable.
ProoJ Suppose j g, 1 = 1, and 9) is not badly approximable, It suffices to show that ind(p) > 0. For this, choose g E A(D) such that jj rp - g i/ < ~! q lj = 1. Then /I 1 - qg II < 1, so that qg is an exponential, and q and g have the same index. Since the index of an analytic function is nonnegative, ind(y) > 0. QED.
In Section 2, we give an elementary proof of the remaining implication of Poreda’s theorem.
A point z E T is an A(D)-essential boundary pokt of D if for each neigh borhood ofz there exists a function in A(D) which does not extend analytically to that neighborhood. The A(D)-essential boundary points form a closed subset of I’ which includes the boundary of the complement of D.
In Siection 3, a simple duality argument is used to prove the following.
THEOREM 1.2. Each badly approximable firnction in C(F) has constant mod&s 011 the set of A(D)-essential boundary poirlts 03f D.
Theorem 1.2 reduces questions about badly approximable functions to the unimodular case. It turns out (Section 6) that for certain domains D, there are unimodular functions in C(r) with arbitrariiy large winding numbers, which are still badly approximabie. Our principal result is the following partial converse the Theorem 1.1.
TXEOREM 1.3. Suppose that r comists of N + I disjoint closed Jordaiz cuves. Ifs, E C(T) is badly approximable, then y has mm321'0 constant modt&.u, and
ind(qs) < hr.

282 GAMELIN ef a/.
Theorems 1.1 and 1.3 include Poreda’s theorem, which corresponds to the case N = 0. One proof of Theorem 1.3, using the dual extremal method, is given in Sections 4 and 5. A second proof, using Toeplitz operators, is given in Section 7. An example given in Section 6 shows that the range 0 < ind(v) < N is indeterminate. Finally, in Section 8 we extend the results to finite Riemann surfaces.
2. AN ELEMENTARY PROOF OF POREDA'S THEOREM
It suffices to consider the case in which D is the open unit disc A. Let y E C(r) satisfy /I y 11 = 1. In view of Theorem 1.1, it suffices to show that either of the conditions
97 is not unimodular, (2.1)
q is unimodular and ind(y) > 0, (2.2)
implies that d(y, A(A)) < 1. Suppose that (2.1) is valid. Choose b < 1 so near to 1 that the set E =
{IV E I’ : b < 1 v(w)1 < l> is a proper subset of r. Then E is simply connected, so that arg(pj) has a continuous determination on E. Consequently there is a smooth function v E C,(r) such that I arg(y) - 2’ / < ~r/4 on E. The harmonic conjugate *v of v is then continuous on r [14], and g = exp(iv - *v) belongs to A(A). The range of g/y on E is contained in the sector (I arg z [ < n/4], so that for 6 > 0 sufficiently small, the range of 6g/p, on E is contained in the open disc centered at 1 with radius 1. Hence
on E. If 6 > 0 is small, also j v - Sg / < 1 on r\E, so that II v - ag jl < 1, and d(y, A(A)) < 1.
Next suppose that (2.2) is valid, and set m = ind(y). Write y = z”ki”, where II E C,(r). Let u E C,(r) be a smooth function which satisfies jl u - u 11 < r/4. As before, set g = exp(iv - v*) E A(A). Again the range of Gge-iU is contained in the open disc centered at 1 with radius 1, for 6 > 0 sufficiently small. Consequently
a!(~,, A(A)) < 11 v - hFJzg\j = jj 1 - Sge-iU I/ < 1.
This completes the proof.

ON BADLY AF’PROXIMABLE FUNeTIONS 283
3. DUAL EXTREMAL MEASURES
L2t A(D)’ denote the (finite regular Borei) measures on F which are orthogonal to A(D). By the Hahn-Banach theorem, there is for each q E C(r) a measure ,U E A(D)’ such that 11 p jl = 1 and
If q is badly approximable, then the chain of inequalities 1; 9 /, = d(o, A(D)) = Sip cI,, < J 1 ~JJ / cl 1 1~ j < I/ g, Ij become all equalities. We conclude that
w a 0 (3,1)
! 9) 1 = /I e- I! on the closed support of,u. j3.2j
Conversely, if there is a nonzero measure p E A(D)’ for which (3.1) and (3.2) a.re valid, then
so that 9 is badly approximable. Any nonzero measure p E A(QL satisfying (3.1) and (3.2) is called a EElinl
~~t~nzal nzeaswe for 9. Then g, E C(r) is badly approximable if and only if there is a dual extremal measure for g?. Theorem 1.2 is now an immediate consequence of (3.2) and the following lemma.
kEMMA 3. 10 If p is a 12onzero measure in A(D)‘, then the ciosed swp~~rt C$ y contafns the A(D)-essential boundary points of D.
ProoJ We will use some facts about the Cauchy transform + of a measure T on r, defined by
The integral converges absolutely for almost ali (do &) complex numbers -7:, and i is analytic off the closed support of T. If-i = 0 a.e. (& Sjs then r = 0. Finally, if 7 E A(D then + = 0 a.e. (do &) on the complement of D, SO that T is completely determined by the analytic function -i on D I-?: Lemma 1.11.
Now let p be a nonzero measure in A(D)‘, and suppose H,, E F dces not he in the closed support of p. Choose 6 > 0 so that the disc Lf, = (1 z - -70 / < S\, carries no mass for p. Then @ is analytic on 8, , and ,L is not identically zero on D. Hence $ vanishes on no open subset of A, . and d, C 19;

284 GAMELIN et d.
LetfE A(D). For fixed z E D, the function [f(z) - f(?J]/(z - c), regarded as a function of 5, belongs to A(D). Hence
s z E D. Solving for f(z), we obtain
z E D.
This formula shows thatfextends meromorphically to d, . The meromorphic extension must coincide with the continuous extension off from D to D, so that f is analytic on d, . Consequently z,, is not an A(D)-essential boundary point of D. That proves the lemma.
On the basis of Lemma 3.1 it is easy to see that the set of A(D)-essential boundary points of D coincides with the Shilov boundary of A(D).
4. SOME PREPARATORY LEM~L~S
For p > 0, the space P(V) associated with a domain V consists of the analytic functions f on V such that j f j ZJ has a harmonic majorant. If J is an analytic arc which forms a relatively open subset of aV, then the nontangen- tial boundary values of such an f exist almost everywhere with respect to the arc length measure on J. The boundary value function will also be denoted byf.
As usual, the open unit disc will be denoted by d. A theorem of Helson and Sarason [6] and Neuwirth and Newman [9] asserts that if f E lW(A) has positive radial boundary values a.e.(dO) on ad, then f is constant. The main idea of their proofs also serves to establish the following local version, which is due to Koosis [7].
LEMMA 4.1. Let f E Hll”(A), and let J be an open arc on ZA. If the radial boundary values off are positive a.e. (de) on J, then f extends analytically across J.
Since the proof is brief, we include it. Write f = BF”, where B is a Blaschke product and FE H’(A). The condition that f(z) > 0 on J becomes the condi- tion B(z) F(z) = F(l/Z) on J. The result of the lemma now follows from the H1 version of Morera’s theorem (cf. [l l]), which shows that if g E Hz(A) agrees on an arc J of aA with a function G E H1({/ z 1 > I)), then g extends analytically across J.

op\: BADLY APPROXIMABLE FUNCTIONS 285
A conformaily invariant statement of Lemma 4.1 is as follows.
LEMhIA 4.2. Let Y be a domain, and let J be an onnll’tic arc which forms nn open subset of 8V. Suppose f E H1p( V) satisjes fdz 3 0 along J. Then f extends annni~~tically across J.
In the following Iemma, we do not know whether the E can be taken to be 0.
LEMMA 4.3. Let V be a domain, and let J be cm nnalqk arc which form at open subset of EV. Let E > 0, and let f E HE+l/s( / J ). If there is c1 continr~oz~~ Inz~~lzodl[~.rlurfirrlctiolz y on J such that yf dz 3 0 along J, then f is of class 15~2, for allp < co, near each compact subarc of J,
Proof. The problem is local, so that we can assume that V = d. Let I be a relatively compact subarc of J, and let ZI E C,(Ulj satisfy y = SU near 1. [14, Chap. VII, Theorem 2.11(ii)], exp(iu - 51) is of class HP for allp < a. Hence g = exp(iu - u*)f~ H1/2(d). Furthermore, gdz > 0 along 1. By Lemma 4.2, g extends analytically across 1. Since exp(--izi +- *zi) also is of class ND for all finite y, f is of class H” for all p < a, near compact subsets of 1.
The following lemma is a standard variant of the argument principle [8, Chap. III, Sec. lo].
bhlMA 4.4. Suppose that the boundary p of 1) consists of N $- i simpk closed analytic Jordan curves. Suppose f is nreromorphic on a neighborhood of B: and arg( fdz) is colzstant on each compotzent of T* Thez the difference of hk nttmber of zeros off afld the number of poles off on D is N - 1. (Here the zeros or poles gff on rare cozmted according to half their i?mlt@icity.)
5. PROOF OF THEoRm 1.4
To prove Theorem 1.3, we can and will assume that the boundary F of D consists of N + 1 simple closed analytic Jordan curves. In this case, the measures p E A( are precisely the measures of the form p = fdz, when f E HI(D) (cf. [ 111). A dual extremal measure will be referred to as a && extwmnE diffeerentiaI. Let y be a unimodular function in C(T). Denote by rO the ‘“outside” component of T, and by r, ,...j T, the ‘“inside” components. Let ~j be any fixed point inside I’, , 1 < j < N, and let zG E D. For appro- priate integers m, )..., 3~2~~ , we can express
pi(z) = [(z - z,)/j z - z. I]’ dmm) ei”g(z)/j g(z)1 .
where z’ E C,(T), and
(5.1)
g(z) = (z - ZJ’l ... (z - z;JQ

286 GAMELIN et d.
is an invertible function in A(D). Define Uj E C,(r) to be 1 on r? and 0 on I’\I’i , 1 <.j < h? Then there are constants c1 ,..., cN such that
U = 1' - 1 CjUj
has a single-valued harmonic conjugate function *II on D [8, Chap. I, Sect. lo]. Define ~~ = exp(i C cju,), so that
90 = 1 on To, = exp(ic,) onYj, 1 <.i<N.
The formula (5.1) becomes
(5.2)
q = ~~[(z - zo)/l z - z, [lindcm) eiug(z)/j g(z)[ . (5.3)
Now suppose that y is badly approximable, and thatfdz E A(D)l is a dual extremal differential. By Lemma 4.3, f is of class HP on D, for all p < co. Consequently the function
G = fg exp(izd - *u) (5.4)
belongs to HI’ for all p < co. The relation &dz 2 0 becomes
yo(z - ~~)~nd(e) Gdz 3 0 along r. (5.5)
By Lemma 4.2, G extends analytically across r. Furthermore, the mero- morphic differential (z - zo)ind(m) Gdz has constant argument along each component of r. From Lemma 4.4, we conclude that G has N - 1 - ind(q?) zeros on 4, where the zeros of G on r are counted according to half their multiplicity. Setting F = G/g, we obtain the following.
THEOREM 5.1. Let g, be a z&modular function in C(r) which is badly approximable, and let fdz be a nonzer’o dual extremal differential for q~ Then there are zz E C,(r) and an analytic fzrnction F on B, such that u has a single- ualued hasmonic conjugate *zz, and
f(z) = F(z) exp(-iu + *u).
Ftlrthemzore, F has N - 1 - ind(F) zeros on ii, where the zeros of F on r are cozmted according to half their multiplicity.
Theorem 1.3 is an immediate consequence of Theorem 5.1. Indeed, since the number of zeros of F cannot be negative, we obtain from Theorem 5.1 the estimate
ind(F) < N - 1
whenever y is badly approximable.

ON BADLY APPROXIMABLE FUNCTIONS 287
Now return to the formula (5.3), and suppose that ind(q) = 0. If q is badly approximable, with dual extremal differentialfdz, then (5.4) and (5.5) show that y0 is also badly approximable. Conversely, if ?0 is badly approximable. with dual extremal differential G, then the H1 function f defined by (5.4) satisfies q$dz 3 0, so that 91 is badly approximable. We conclude that g, and e,, are simultaneously badly approximable or not, when ind(q) = 0.
Unfortunately we do not know which locally constant functions q+, are badly approximabie. At first we suspected that a locally constant unimodular function q?O is not badly approximable if and only if its range lies on a subarc of ?d of length less than rr. An example given in the next section shows that this guess fails. The following trivial observation, valid for arbitrary domains, is sufficient to lead to complete information in the case of an annulus.
LEE&IA 5.2. Let y0 be a continuous unimodular fimctiorr on %I3 which assumes only two caiues. Then y0 is badly approximabk iJ’and on/y ty the two aaluer are dim~etrically opposite each other.
PFOOJ If the values of y, are not diametrically opposed, then their average g is a constant function which satisned 11 y0 -- g 1; < I, so that 4p0 is not badly approximable. On the other hand, if the values are diametrically opposed, then no lz E A(D) can satisfy jj q0 - h Ij < 1, or else there would be a line passing through 0 and separating the range of h on r, from the range of P% on F, : an absurdity.
THEOREM 5.3. Fix 0 < B < 1, and let D be the annulus (r < / z j < 1). Let y be a wukoduiar function in C(r). Therz p is badly approximble if and o~;ly [feiflrer ind(g?) < 0, or ind(q) = 0 and
[Here we Megrate continuous determinations of arg(yj on the rerpectice iMerual.s of ktegration .]
PlooJ In view of Theorems 1.1 and 1.3, it s&ices to consider the case md(,-) = 0. With rl = 0, the formula (5.2) then becomes
p?(z) = z’neiLA, /,71 =I = (zm/r”) eiueie, /zl =r.
Moreover, F, = 1 on r,, and F,, = eic on r, . By Lemma 5.2 and the remarks preceding that lemma, y is badly approximable if and only if y,, is, and this occurs if and only if c m= 7~ (mod 27~). Now u + I’*u is analytic, so that JTV Ir(eis> d8 = JTz u(rei”) dtI, and the left-hand side of (5.6) is computed to be --(mod 2~). This completes the proof.

288 GAMELIN et al.
6. SOME EXAMPLES
First we show that the estimate of Theorem 3.1 is sharp, for any D. It suffices to consider the case in which the N + 1 Jordan curves which form the boundary r of D are analytic. In this case, set
when ds is the arc length measure on r. Then g, is continuous and unimodular on r. Since the argument of dz/ds increases by 2~ around the “outside” contour of I’, and it decreases by 29-r around each of the N “inside” contours of r, the index of dzlds is 1 - N, and
ind(v) = N - 1.
Since g, dz > 0, dz is a dual extremal differential for cp, and cp is badly approximable.
More generally, for any integer k > 0, there is a badly approximable function on i3D with index N - 1 - k. Indeed, for fixed z0 E D, the function
has index N - 1 - k. Since it has the dual extremal differential (z - z,,)” dz, it is badly approximable.
The remaining examples depend on the following lemma.
LEMMA 6.1. Suppose that r consists of N + 1 disjoint circles I’, , r, ,..., r, , where I’, is the “outside” boundary circle. Suppose also that all the boundary circles are centered on the real axis R. Let tI ,..., tNmI be points qf D n R, such that between each two consecutive ‘inside” circles these lies exactly one of the tj’s. Then there is a nonzero analytic d$feerential fdz on B such that
fdz < 0 012 r, , fdz > 0 onPi, <j<N,
f (tJ = 0, 1 <j<N- 1.
Proof. We map D conformally onto a slit domain V obtained from the complex plane by excising slits (- co, 01, [a, , b,] ,..., [aN , bX] along the real axis, so that r,, corresponds to the slit (-co, 01. Let wj be the image of tj . Then between each pair of consecutive bounded slits there lies exactly one of the wi’s.

ON BADLY APPROXIMABLE FUNCTIONS 289
Define an analytic differential o on V by
(IV - WI) ... (w - w& hl, Co = i[W(lV - a,)(1v - 6,) . . . (,,' - (&&,I? - bN)]l/? '
where the branch of the square root is chosen to be positive for large positive values of ~1 (cf. [13, p. 2931). One checks that w > 0 along the respective sides of the bounded slits, while w < 0 along the sides of the slit (-co, 01. Tine pulIbackf& of OJ to D has the properties asserted by the lemma.
Now let D and the t,‘s be as above. Define F,, = - 1 on r, and q0 = I on Fir,, . If 4;1z is the differential of the lemma, then q, fdz 3 0 along IT Fix an integer i?z satisfying
and define
z E r.
4)
(6.1)
Then
ind(F,) = IH.
On the other hand, the analytic function
-
satisfies qlng& > 0, so that yli( is badly approximable. This shows again that the estimate of Theorem 1.3 cannot be improved upon
Now consider an infinitely connected domain W obtained from the open unit disc d by excising the origin (0) together with a sequence of disjoint ciosed subdiscs (AJjm=l , whose centers (c,.} lie on the positive real axis and decrease to 0. We claim that for each integer 112, there is a badly approxi- mabie function vVz on a W with index IX. Indeed, let (tj>;“=l be a sequence in tV .n R such that tj lies between dj and dj+, . Define y0 to be - 1 on cd and + i on (3 W)\(aL?), and define qjm as in (6.1). If D, = d\uTl, dj : then the preceding work shows that
n’(%n Ii+ A(D,v)) = 1, h: 2 I??.
By [S, p. 521, U&D,) is a dense subspace of A(W). Consequently thedistances n(A I?~,~, A(D,)) decrease to d(lz, A(W)) whenever 11 E C(a W). In particular, d(vh 3 A(W)) = [I yrn j/ , so that y,,, is a badly approximabie function with index m.

290 GAMELIN et al.
The final example is that of a locally constant unimodular function ‘p whose range lies on no arc of ad of length less than 7~, but such that F is not badly approximable. For this we take D to be a circle domain as in Lemma 6.1 with only three boundary circles (N = 2) such that D is symmetric with respect to the imaginary axis. In other words, r, is centered at 0, r, and I’Z have equal radii, and the center of J-‘, is the negative of the center of r, . Set v = 1 on I’,, , y = i on r, , and y = -i on r, . We claim that 4~3 4DN < 1.
Indeed, suppose that q is badly approximable. Let f(z) dz be a dual extremal differential for v. Then f is not identically zero, and &dz 3 0.
Since ‘p( -?) = F(Z), also v(z)f(--Z) dz > 0. Furthermore the inequality yfdz < q[f(z) + f(-Z)] dz shows that f(z) + f(-2) is not identically zero. Replacingf by f(z) + f(-5)>, we can assume that
f(z) = f(-a, z E 0.
Let z,, be the zero of J Since f has only a single zero, (6.2) shows that z, = -5, 3 and thus z,, lies on the imaginary axis.
According to Lemma 6.1, there is a nonzero analytic differential g(z) dz on D such that g(z) dz < 0 along r, , g(z) dz >, 0 along .P, u rZ, and g(0) = 0. Set h = f/g. If z0 = 0 then h is a bounded analytic function whose argument assumes distinct constant values on the components of r, an absurdity. We conclude that z, # 0. Consequently h is meromorphic, h has a simple pole at 0, and lz has a simple zero at z,, (a double zero, if z0 E r). Moreover, h maps D conformally onto a slit domain Won the Riemann sphere. If S, , S, , and Se are the slits that correspond respectively to r, , r, , and r, , then S, _C (- co, 01, S1 _C (i0, ia), and S, C (--iO, -ico). Now replacing g by g(z) + g(-Z), we can also assume that g satisfies the same functional relation (6.2) as J
Then also h(s) = h(-5). In other words, the reflection z + --Z of D in the imaginary axis corresponds via h to the reflection w -+ 3 of the slit domain W.
Now let Z/J be the conformal self-map of W which is induced by the con- formal map z + -z of D, that is, #(w) = h(--h-l(w)). Then # leaves the real axis invariant, # interchanges the upper and lower half-planes, #(co) = XI, and 4(O) + 0. The map 1~ + #(MT) then yields an anticonformal self-map 4 of the slit upper half-plane H+\S, . Now H+\S, is conformally an annulus, and any anticonformal self-map of an annulus is a reflection, which is completely determined by specifying a fixed-point on the boundary. We conclude that $ must be the anticonformal map w + - w of H+\S, because both leave 03 fixed. However w + --W leaves 0 fixed, whereas 4 does not. This contradiction establishes the assertion.
We remark that in the case of a circle domain D with three boundary

ON BADLY APPROXIMAJSLE FUNCTIONS 291
circles, there is a close relation between the family of locally constant uni- modular functions which are badly approximable and the family of con- formal maps of D onto radial slit domains on the Riemann sphere. It turns out that the precise description of the badly approximable locally constant functions depends on the size and configuration of the boundary circles of L3.
7. TOEPLITZ OPERATORS
In this section, we indicate the connection between the dual extremai problems under consideration and certain Toeplitz operators. This will lead to another simple proof of Poreda’s theorem, and an alternative proof of Theorem 1.3. For details on Toeplitz operators, see [4]? which is the source for some of the proofs in this section.
Let T be a positive kite measure on r, and let M be a closed subspace of I.‘(T) such that
A(D) MC M. (7,l)
For fixed q, E D, the operator f + (z - zO)f, f E MY has closed range and null space (01. By the theory of Fredholm operators, the range (r - zO) M of these operators has the same codimension (finite or infinite) in M for all q E D. We will be interested in the case that
(Z - q,) M has codimension one in M, for Z, E D. (7.2)
Let P be the orthogonal projection of L’(r) onto M. For each 4~ E k”(r), the Toeplitz operator T, is defined on A4 by
Tat- = PC&f), f E iv.
The correspondence v -+ T, is a contractive linear mapping from L=(r) to the bounded operators on M, which satisfies TO* = T, and TI = 1~ If i” E L”(T) and # E A(D), T,, = T,T$, and T,,J = TJ, .
LEMN.~ 7.1. Suppose that (7.1) and (7.2) at’e uaiid. Then T,T, - TQti is a compact operator whenever gj, # E C(r). Furthermore, if y E C(r) does not vanish on S, then T, is a Fredholm operator, and
ind(y) = -index(T,). (7.3)
Proof= Here
index(T,) = dim M(T,) - cod ZjT,)

292 CAMELIN et a/.
where 9? denotes “range” and J- denotes “null space.” Now T,T$ - T,, = 0 when # E A(D). If #(z) = l/(~ - z,,) for some fixed z0 ED, then T,T, = T,, on (z - z,,) M, so that T,T, - Tm$ is one-dimensional, hence compact. Since linear combinations of functions in A(D) and the functions l/(z - zJ, z,, ED, are dense in C(r) [3], TojT, - T,, is compact for all y, # E C(r).
Suppose v E C(r) does not vanish anywhere on l? For z,, E D fixed, we can express
q(z) = (z - z(p gh,
where m = ind(v), g is an invertible function in A(D), and h E C(r) has a continuous logarithm on r. Then h is appropriately homotopic to the con- stant function 1, so that Tn has index zero. Since T, is invertible, its index is zero. By (7.2) the index of the Toeplitz operator of (Z - z,,),, is --r?z. Con- sequently index (T,) = --m = -ind(gj). Q.E.D.
The usual Toeplitz operators are obtained by setting D equal to the open unit disc d, and setting M = Hz(&). In this case, Poreda’s theorem can be proved as follows. Assume g, E C(i3D) is unimodular. Then d(v, A@)) < 1 if and only if T, is left invertible, that is, if and only if dim J”(T,) > 0 [4, p. 1871. If then g, is badly approximable, we have dim JV(T,) > 0. By Coburn’s lemma [4, p. 1851, cod %(T,) = 0 so (7.3) shows that ind(F) < 0. On the other hand, if y is not badly approximable, then dim N(T,) = 0 and (7.3) yields ind(gj) 3 0, which does it.
To extend this proof, we require an analog of Coburn’s lemma, and a criterion relating the distance estimate to left invertibility. A criterion suffi- cient for our purposes can be found in the work of Abrahamse [l]. The precise fact we will need can be proved for infinitely connected domains. It is the following.
LEMMA 7.2. Let F E C(r) b e uninzodular. Then p? is badly approximable if and only if there are a positive measure T on r and a subspace M of L2(7) satisfying (7.1) and (7.3, such tlzat the Toeplitz operator T, on M is not left imertible, that is, N(T,) + (0).
Proof. If 9) is not badly approximable, there is g E A(D) satisfying 11 g - q // < 1. Since q is unimodular, II 1 - gg, II < 1. Hence Ij TIpem I/ = /I I - TgT, I/ < 1, so that TeTw is invertible, and T, is left invertible.
On the other hand, suppose that q is badly approximable. Let p E A(D)l be a dual extremal measure for 9, so that q+ > 0. Let T = pp, and let M = h!“(T) be the closure of A(D) in L”(7). If g E A(D), then sg+dT = sgdp = 0, so that g, _L H’(T). From the definition of T, , we obtain T,(l) = 0, and 1 E J(T,). It suffices now to establish (7.2).
Suppose that (z - z,,) M = M. Then l/(z - z,,)(n E M for all integers

ON BADLY APPROXIMABLE FUNCTIONS 293
1~1 3 0. Hence s (l/(z - zO)nlj dp = 0 for all m > I. Hence p is orthogonal to the linear span of the functions in A(D) and the l/(z - QizY YIZ > 1 [3]. Since this linear span is dense in C(F), we obtain p = 0, a contradiction.
It follows that the closed subspace (z - zO) M of M has codimension at least 1 in M. Since (z - z,,) A(D) has codimension 1 in A(D), (z - zOj M has codimension precisely 1 in AI, and (7.2) is valid.
The required analog of Coburn’s Lemma is as fo!lows,
LEMMA 7.3. Suppose that r consists of N + P disjoint simple closed analytic Jordan c1oze.s. Let r be a posititle measure on r! which is absolutely continuous with respect to the arc length measure on r. 4, q E L”(r) satisfies JV(T,) * {O], then
dim Jlr(T,) < N.
ProofI Letf~~lr(T,),ff 0, Then
1 c&la dr = 0, all h E M. (7.4) ”
In particular, l F j .f 1% $ d7 = 0 for all # E A(D). Consequently @ j f jB d7 is an analytic differential of class W. It follows that T is mutually absomtely continuous with respect to arc length ds, and thatfcannot vanish on a set of positive measure. -
Now iet g E X(T,). Then j g? gl;dT = 0 for all h E AI, SC that S q gf$dT = 0 for ail Z/J E A(D). Setting h = +g in (7.4), we find also that S q gV@r& = 0 for all $ E A(D). Hence qgfd7 is orthogonal to A(D) + A(D). Since this latter space has defect .N in C(F), and since f cannot vanish on a set of positive measure, the collection of such g’s has dimension at most N.
Alternatiue Proof of Theorem 1.3. We can assume that r consists of Ai + 1 simple closed analytic Jordan curves. Let y E C(r) be a uuimodular badly approximable function. We will show that ind(pj < A?
Take T and M as in Lemma 7.2, so that dV(T,) f (0). Note that the + chosen in Lemma 7.2 is the variation of a measure in A(D)‘, so that in the case at hand, we can assume that T is the modulus of an analytic differential, hence absolutely continuous with respect to arc length measure on -F. By Lemma 7.3, and the relations TF = Tm*: dV(T,*) = B(TcjL, we obtain
cod 9(T,) < 1V.
From Lemma 7.1 we obtain
ind(q) = cod W(T,) - dim A’(T,) < N - 1.
This completes the proof.

294 GAMELIN et al.
Note that the estimate of Lemma 7.3 is sharp. Indeed, Section 6 provides a circle domain D bounded by N + 1 circles, and a unimodular 9) E C(r) such that v is badly approximable, while ind(y) = N - 1. Choosing T and M as in Lemma 7.2, we obtain dim Jlr(T,) = cod W(T,) = ind(p7) + dim Jtr(T,) > N, so that in fact equality must hold. An example in Section 6 also shows that there are infinitely connected domains for which no estimate as in Lemma 7.3 obtains.
8. RIEMANN SURFACES
In this section, we indicate how some of the results of this paper can be extended to Riemann surfaces. Let D be a finite bordered Riemann surface with interior genus P, such that the boundary I’ of D consists of N + 1 closed analytic curves. Again A(D) is the algebra of analytic functions on D which extend continuously to I’, and A( consists of measures on P which are the boundary values of analytic differentials on D of class H”. The proof of Theorem 1.1 is valid in this context. The analogue of Theorem 1.3 is the following.
THEOREM 8.1. If g, E C(r) is badly approximable, then v has nonzero constant modulus, and
ind(y) < 2P + N.
The theory of Toeplitz operators developed in Section 7 also carries over to this context. Fix a function F analytic on B such that F has only one zero on B, a simple zero at some point of D. Let T be a finite measure on I’, and let A4 be a closed subspace of L”(7) such that
A(D) MC M, (8.1)
FM has codimension one in M. (8.2)
The Toeplitz operators T, on M are defined as before, and Lemma 7.1 is valid. The proof of Lemma 7.1 also carries over to this context, once one makes the following two observations: First, the linear span of A(D) and the functions l/F”, m 3 1, is dense in C(r) [I I]. Secondly, if v is a nonvanishing function on r with index m, then there are lz E C,(r) and an invertible function g E A(D) such that y = F”‘g exp(lz).
The proof of Lemma 7.2 also carries over, once one replaces z - z0 by F. Lemma 7.3 is also valid, except that one obtains only
JV( T,) f {0} implies dim J-(7’,,) < 2P + N, (8.3)

ON BADLY APPROXIMABLE FUNCTIONS 295
because A(D) + ,4(D) has defect 2P + N in C(r) [I l]. The alternative proof of Theorem I.3 given in Section 7 then serves to establish the estimate given in Theorem 8.1.
Again the estimates of Theorem 8.1 and the analog (8.3) of Coburn’s lemma are sharp. To see this, we proceed as follows
Let a be any analytic differential on D which has no zeros, and let 7 Se the measure on r defined by T = / 01 1 . Then Q- = qq where 9 is a continuous unimodular function on r. Furthermore, r is a dual extremal differential for p;, so that F is badly approximable. Let M be the closure of A(D) in L”(T), and consider the Toeplitz operator T, on M. Since
0 = J’ga = j’gqidr,
the projection of y1 into M is 0, and
T,(l) = 0.
Let w be a Schottky differential for D, that is, w is an analytic differentiai on D which is real along r {cf. [ll]). Then w/o: = h is analytic on D. Moreover, if g E A(D), then
o- [@o- “r
f g/W = [@gdT. ‘i-
It follows that h E N(T,). Since the dimension of the space of Schottky differentials is 2P + N, the dimension of A”(T,> is at least 2P + IV, so that from (8.3) its dimension is precisely 2P + N, and in particular the estimate (8.3) is sharp. One checks that only the constants lie in N(T,), so that dim A’-(T,) = 1, and
ind(F) = -index(T,) = 2P + i’J - 1.
Hence the estimate of Theorem 8.1 is also sharp.
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