Annals of Pure and Applied Logic 63 (1993) 283-297

North-Holland

283

Infinitesimal approach automorphic functions

of almost-

Yves Pkraire Dkpartement de Mathkmatiques, Uniuersite Blaire Pascal (Clermont II), B. P. 45, 63170 AubiPre, France

Communicated by J.Y. Girard

Received 28 February 1992

Revised 25 February 1993

Abstract

Peraire, Y., Infinitesimal approach to almost-automorphic functions, Annals of Pure and

Applied Logic 63 (1993) 283-297.

Thanks to the use of ideal elements of several levels, we are able to give a compact topological

characterization of almost-automorphic functions. This new characterization turns out to be

equivalent to a geometrical one: the existence of a relatively dense group of “pointwise

periods”. However, the more significant result obtained, in our opinion, is a very important

lowering of the complexity in characterizations and proofs.

Introduction

Almost-automorphy is a property of almost-periodic functions which was used by Bochner in [l]. A real-valued function on the group Z of integers is almost-automorphic if it has the property that from any sequence {a,} c 72 we can extract a subsequence {b,} such that a function g exists with both lim,_mf(t + b,) =g(t) and lim,,,g(t - b,) =f(t) for any t E Z. In [l] Bochner said he did not know if almost-automorphic functions were genuinely more general than almost-periodic functions. However in [5], Veech has observed that an almost-automorphic function is not necessarily almost-periodic: for example, the function (cited in [5]) defined on Z by f(t) = sgn(sin(t)) is not almost-periodic but almost-automorphic. In [5], Veech has generalized the definition of almost- automorphy to any group G, G being commutative or not. Then he established that his definition, called ‘Bochner almost-automorphy’, was equivalent to another characterization to which he gave the name of ‘Bohr almost- automorphy’.

Correspondence to: Yves Peraire, Dtpartement de Mathematiques, Universite Blaise Pascal (Cler- mont II), B.P. 45, 63170 Aubiere, France.

Elsevier Science Publishers B.V.

284 Y. P&rake

Whenever G is commutative, denoting A(f, F, E) = {p E G 1 Vt E F If(t +p) -

f(t)1 < E}, we may express the Bohr almost-automorphy of the function f by the conjunction of the two following conditions:

(i) for any finite F c G and any real number E > 0, A(f, F, E) is relatively dense;

(ii) for any finite F c G and any real number E > 0, there exists a finite superset F’ of F, and E’ > 0 such that, with additive denotations, the set defined

by

A(f, F’, s’) - A(f, F, c) = {P E G ) p E AU, F’, ~‘1 8~ P $ A(f, F, E)}

is included in A(f, F, E).

Watching the characterization of almost-automorphy according to Bohr, we have the feeling that ‘in an adequate language’, the almost-automorphy of a function can probably be characterized by the existence of a relatively dense group of ‘almost-periods’ (according to the pointwise convergence topology), the former being ideal objects which have not any name in the language of conventional mathematics. Obtaining such a characterization is one among the results of this work. In order to do it, we chose a set-theoretic framework which permits us to deal with objects of several ideality levels. This set theory is a conservative extension of the Zermelo-Frxnkel set theory with the axiom of choice (ZFC). Besides the fact that we have obtained a more important power of saying, we have achieved a really very important lowering in formulations and proofs which overcomes our expectations.

1. Some tools from relative analysis

The set-theoretic framework is a conservative extension of ZFC. In addition to the usual undefined predicate E, the language of our theory involves finitely (intuitively) many new undefined predicates (unary predicate), ‘id, ‘id, . . . , vid, where N is a fixed large enough integer (in this paper N = 4 suffices). The formula ‘“id(x)’ is to be read ‘x is “ideal’.

The first axiom is:

Including axiom. For any p <N we have: V,X (Pid(x) 3 P+Lid(~)).

If a set is ‘ideal we shall say also that this set is standard.

We adopt the following abbreviations:

Vpx F(x) for Vx [Pid(x) 3 F(x)],

ilpx F(x) for 3x [Pid(X) & F(x)].

We say that (ml, m2, . . . , mk) is a multi-level of ideality if each mi is either, 1, 2, . . * ) N, or the symbol ‘co’. We agree to write: ~0 > N.

Infinitesimal approach of almost-automorphic functions 285

If (m,, m2,. . . , mk) and (n,, nz, . . . , nk) are multi-levels we say that

( m,, m2,. . . , mk) dominates (n,, n2, . . . , nk), if we have jointly:

(a) min(m,) 2 min(n,),

(b) foranyi,jE{1,2 ,... , k} we have: m,<mj e n;<nj.

Example. (4,3,2, ~0) dominates (3,2,1,5).

A formula F is said to be *external if not predicate “id with p < II, occurs in it.

We call a formula internal in case it does not involve any predicate Pid.

If we replace, in the formula F, each subformula of the form ‘nid(x)’ by the

equivalent one 3”~ (y =x), if we note V” for V and 3” for 3, then we write

F = @(nl,‘zz,---.nr) to indicate that the internal or external quantifiers, V”,, or 3”l

occur in the formula equivalent to F so obtained.

Example. We denote:

@‘J,=) = 3s E R V2t E R 3’F > 0 3’u E R (If(s - t + u) -f(u)1 > E).

The other extra-axioms are as follows.

Principle of transfer (T). If @(“1*“2~--.~“k) is an “external formula with all the

parameters occurring in it “ideal, and if (m,, m2, . . . , mk) is a multi-level such

that (ml, m2, . . . , mk) dominates (n,, n2, . . . , n,), then

@’ n,.nz ,.... nk) e @m,,m, . . . . . m*).

Example. If the constant f is a standard function, the next equivalence holds:

~(1.2.~) e gj(2.4.5) = gSs E R v4t E [w i12.E > 0 g2u E R (If (s - t + CL) -f (u)l > E).

Corollary. If F(x) is an internal formula, then for any “ideal value of the parameters we have

Vx F(x) e Vx F(x).

As a consequence, every set defined with unicity in ZFC is ‘ideal (standard); so

we have ‘id(O), ‘id(N), ‘id(R), etc.

Principle of idealization (I). If @(x, y) is an internal formula with free variables x

and y and possibly other free variables, if p < q, q may be 00, then we have:

[W”“F Ty Vx E F @(x, y)] e Py W’x E F @(x, y).

We shall make use many times of the following consequence of the idealization

principle.

If p <q, q may be CD, then there exists an Yidealfinite set F, which contains all

the “ideal sets.

286 Y. Phaire

Principle of standardization. If Q(X) is an “external formula, with the free variable x and possibly other free variables, then

V”z Yy V”x [X E y e (x E z & Q(x))].

If z is a fixed “ideal set, the set y, the existence of which is asserted by the former principle, is unique (use (T)). We say that y is the “standardized of the ‘external set’ (it is not an object of our set theory): ‘{x E z; a(x)}‘. We denote y = “st{x E 2: G(x)}.

In the cases where @p(x) is internal and all the parameter standard, we deduce from (T) that y = {x E z: Q(x)}. Hence, our principle of external comprehension is a generalization of the principle of comprehension of ZFC.

From now on, if x, y are two real numbers and f, g two elements of the set RG of the functions from G to I%!, we write: l x “= y if x is “infinitely close to y relatively to the usual topology of R.

It means that, for any “ideal number E > 0, we have Ix - y I< E.

n m m e y IwdfE=giffis (’ fi ‘t 1 close) to g relatively to the pointwise topology on

It means that, for any “ideal element t of G, we have f(t) “= g(t).

l f “= g if f is “(infinitely close) to g relatively to the uniform topology on RG. It means that, for any element t of G, we have f(t) “= g(t).

This theory and its applications to analysis are studied in detail in [4]. However, we think that an intuitive approach from the previous element is sufficient to understand the sequel. Intuitively, a topology being given, the sentence ‘x is 2infinitely close to y’ may be understood as ‘x is infinitely close to y’ (according to the given topology). However, it can as well be understood as ‘x is very very close to y’.

For reasons of convenience, although this is not really necessary, we shall introduce a notion of “shadow for each of the three considered topologies.

We say that the real number a is the “shadow of the real number x if “id(a) and x “= a. We denote: a =” x.

Similarly, if f and g are functions from G to R, we say that g is the uniform ‘%hadow (respectively, the pointwise “shadow) of f if “id(g) and f “= g (respectively f “= g). We respectively denote:

“f = the uniform “shadow off,

“f = the pointwise “shadow off.

It proved to be convenient to extend the three kinds of ‘functions’, x+ “x, f+ “f and f + “f, as follows.

(i) We fix an “‘(infinitely large) real number #, (v(infinitely large) means larger than any positive “‘ideal number) and we denote also by # the constant function, x -+ #.

(ii) We let for n s N, no = # if the real number x is not “(infinitely close) to some “ideal number, “f = # (resp. “f = #), if the function f is not “(infinitely

Infinitesimal approach of almost-automorphic functions 287

close) to some “ideal function, relatively to the uniform (resp. pointwise) topology.

It is easy to verify that for any iz s N, we have:

“#=#, # being the number or the function,

“# = #.

If f E [wG and CT E G we call the element of RG defined by

for any t E G, fO(t) =f(o + t)

the a-translate fC off. For the sake of legibility, we consider that in an expression involving

translations and passages to shadow, we begin by the translations. So, for example, we shall write:

% for “(fa), VA for “CY_L)J.

Rules to handle the shadows

For any n c N we have:

Rule 1. “# = #.

Rule 2. “# = “# = #.

Rule 3. If x E [w then (3 = # e 1x1 is “(infinitely large).

Rule 4. If “f # # then “f = “$

Rule 5. “f # # e Vpt “f (t) # St.

Rule 6. If 1 c n cm s N, if “id(f) and “f # #, then “f # # (and therefore “f = “f according to Rule 4) if and only if tl”t .s G “(f (t)) = “([“f J(t)).

Rule 7. If 1~ II < m 6 N, “f # # and “f # #, then “(“f) = “f.

Rule 8. If If ( is bounded by a number M such that “id(M), n <N, then “f # #.

Rule9. Ifl~n~m~N,“~##and~f#f,then”(‘~~)=~~

Rule 10. If If1 is bounded and “f =g then, lg/ is bounded (with an “ideal bound).

Rule 11. If 16 yt cm < N and “f f #, then “(“f) = “5

Rule 12. For any (T E G if “f = g # #, then *f. = “go.

288 Y. Ptfraire

Rule 13. V”a E G nfn = (“f),,.

Rule 14. Vu, t E G “(“fO)r = “fO+t

The proof of these rules is left to the reader. We shall need the following external characterizations.

If X is either R with its usual topology or RG, with the pointwise topology, or

the uniform topology, and if A is a standard subset of X then we have: (i) External characterization of the topological closure. A standard element of

X is in the closure of A iff it is ‘(infinitely close) to an element of A. (ii) External characterizations of relative compacity. (a) A is relatively com-

pact iff any element of A is ‘(infinitely close) to a standard element X. (/I) A is relatively compact iff there is a finite subset F of A such that any

element of A is ‘(infinitely close) to an element of F.

The principle of transfer allows us to obtain other characterizations from the former ones.

Example. If M = R! , then condition (CC) of (ii) can be written as follows:

V’A c X (A is relatively compact M (Vx E A ‘x E X)).

Using (T) we can deduce that

V2A c X (A is relatively compact e (Vx E A ‘x E X)), or

V’A c X (A is relatively compact @ (V3x E A *x E A)), etc.

2. Oscillating functions

Notations and definitions

Let H(f) = {fr: t E G}.

Definition 1 (Standard). We say that the function f : G+ R is almost-periodic if H(f) is relatively compact relatively to the topology of uniform convergence.

We denote AP the set of almost-periodic functions from G to R.

We can write, for such a function f with “id(f),

REAP @ VteGcfz## or,

f E AP e Ifi”F c H(f) (Vfr E H(f) 3f0 E F (fr “= fq)).

We have established the next characterization of almost-periodicity that involves the shadows relative to the pointwise topology.

Theorem 1. If f is standard, then f E AP if and only if [Vu, z E G ‘fO # # and

‘fO+T = l(lfoLl.

Infinitesimal approach of almost-automorphic functions 289

Proof. The ‘only if’ part is immediate because f E AP + Vo, t E F ‘f,+,# #;

Rule 14 implies ifO+Z = i(ifD)r and Rules 1, 2 and 4 imply ‘fO+s = ‘(‘fO)t.

Let us prove the converse. We know that for any o E G, if0 # #. We have to

prove that if” = ifg. This arises from the next line:

Vt E G yO(t) = (~~)$.I) ’ = ‘(‘f&(O) = !L+t(O) ‘-fLT+&9 =fu(t). 0

Definition 2. A function f : G -+ R is said almost-automorphic if it is an element

of the set:

AA = ‘st{f E IWG: V~EG l(‘j&, =f}.

Hence we have:

V’f (f E AA e Va E G ‘(lfO)-O = f).

However, by (T), we also have:

Vlf (f E AA (rs V2a E G ‘(‘f&, = f}.

V’f (f E AA e Vo E G ‘(“fO)_, = f},

V’f (f E AA G V”a E G 2(2fU)_0 = f}, etc.

We shall prove in the sequel that the above definition gives the same class of

functions as Bochner’s one.

Remark. It arises immediately from the definition that if k is a real number and if

f and g are two almost-automorphic functions, then f + g, f . g, kf, If 1 and

max(f, g) are almost-automorphic.

Proposition 1. An almost-automorphic function is bounded.

Proof. Suppose f is a standard and unbounded function, then there exists a real

number u such that ‘[f(a)] = #. This implies that If0 = #, (lfU)-O = #, ‘(‘fD)_,, = ‘# = # #f hence, f is not almost-automorphic. 0

Here are other characterizations of almost-automorphy.

Proposition 2. If f E lRG, f is bounded and ‘id(f), then the following conditions (a)(, iy, (c)A(d) and (e) are equivalent:

(:) ‘4’~ E d t/a E G (‘fO = ‘fT 3 ‘fO-, = f ), (c) V*T E G 3a E G (‘fO = ‘fT and lfm-T = f), (d) Vt E G Vo E G Vt’ E G (“fO = *fZ, = ‘fT j ‘fU_r, = f), (e) Vt E G 3a E G g3t’ E G (“fD = ‘fT, = Ifs and ‘f-_,, = f). We may add all the characterizations that can be obtained from the previous

conditions by a change of indices, arising from the principle of transfer.

290 Y. Phaire

Proof. (a) implies (b): V*r (f bounded and standard) + ‘fz # #.

Va ((‘6 = % * ‘&-z= (‘j& = (‘.Q-, + Y2fa-J = ‘& = ‘<%)-5 =f

The last equality expresses the definition of almost-automorphic functions. The others arise from the rules to handle the shadows.

(b) implies (c): V*t ‘& E closure(H(f)). Hence there exists fO E H(f) such that

‘fo = %. For such a u, (b) implies lfO--t = f.

(c) implies (a): L t b e t e an element of G such that *id(r). According to (c) there exists u E G such that (“fO = ‘fr and ‘f,-t = f). For this o we have;

f = ‘f0-r = Y2f0-J = ‘(“fm)-z = l(lfL.

The equivalence of (a), (b) and ( c is now proved. Let us prove that (a), (d) ) and (e) are equivalent.

(a) implies (d): Suppose (a). f E AA implies, according to (b):

V2r’ E G Va E G (‘fO = ‘f%r 3 ‘f,,+ = f)

hence, by (T), we have:

V3t’ E G Vu E G (“fO = ‘ftr + *fg+ = f).

(d) implies (e) is obvious. (e) implies (a): For any t there is an element r’ 3ideal such that 2(‘fs)_s =

‘(lfz)-r and ‘frT = ‘fT for (l(lfs)_T, ‘fT) belongs to the closure of {(1ft)--(7, fO: u E

G}. Likewise, there is a o such that ‘fO = ‘fTs. If (d) is satisfied we have:

f = lfO+ = ‘(3fO_r,) = ‘(yO)_T* = l(lft)_z. = ‘(2(lfJ_r, = ‘(‘(%)-r = ‘(lf&.

Hence, f is almost-automorphic. 0

Definition 3. Let t E G and let f be a standard function from G to R. (i) If we have: Vrt E G f (t + z) ‘= f (t), we say that t is a ‘(almost-period) off.

(ii) If we have: Vt E G f (t + z) ‘= f(t), we say that r is a uniform ‘(almost- period) off.

We denote:

A,(f) = the external set of ‘(almost-periods) off,

P,(f) = the external set of uniform ‘(almost-periods) off.

Applying the principle of transfer to the definition of the ‘(almost-periods) we shall be able to define 2(almost-periods), 3(almost-periods), etc. and the cor-

responding sets A,(f), A3(f ), . . . , Z’*(f), F’3(f), . . . .

Remark. Let f be a ‘ideal function from G to R . The external set Pk(f) is a group. Indeed, if t, z’ E Pk(f), then we have for any t E G:

f(t+t-z’)k=f(t-T’)k= f(t-t’+z’)=f(t).

Infinitesimal approach of almost-automorphic functions 291

Hence t - r’ E P,(f). The analog is not true for Ak(f) because, even if t is ‘ideal, t + z is not generally. However, (Proposition 4) if f is ‘ideal and almost- automorphic, then Ak(f) is a group. This fact is far from being trivial!

Definition 4. An internal or external subset P of G is called relatively dense in G if there exists a finite subset F of G such that G = lJrcF {t} + P.

This definition coincides with the classic one. if P is internal.

Theorem 2. If f is standard then we have: f E AP if and only if P,(f) is relatively

dense in G.

Proof. We shall make use of the criterium @I) for relative compacity. If f is standard then we have: [f almost-periodic] e [H(f) relatively compact] e [there exists a finite subset of RG, {fai, faz, . . . , fan}, such that, for any s E G, there is i(s) E { 1, 2, . . , n} such that fs i= fa,,,3 G [there exists a finite subset,

{ a,, a2,. . . , a,}, of G such that G= iJiGi=n {ai} + P,(f)]. 0

Proposition 3. If f is standard, f E AA, ‘fu E AA and if t is a ‘(almost-period) off and ‘f,, then we have ‘(lfO)r = ‘fOfS.

Proof. Let us denote S = {(fs,ft, fs+J}: (s, t) E G X G}. See Fig. 1. The point (yU, ‘ft, ‘f,+J of (lRG)3 belongs to the closure of the standard set S, relatively to the pointwise topology. Therefore, there are 0’ and t’ such that ‘id(o’, r’) and

(“fO> ‘ftt “f0,++rJ = (‘fO9 ‘fT> !fc7+r). Let us choose o” such that “fO,, = ‘fO. Then we have, f being almost

automorphic:

‘(‘A)-0, = “(%)-~~ =f, (1)

(‘fg)- Oll+S? = (3f0.,)_d’+s’ = ((“f&O& *= “fx, = ‘ft = f. (2)

From (1) and (2) we get

‘(‘fO)L, = l(lfO)-O”+r’. (3)

Fig. 1.

292 Y. Pbaire

Considering ‘fO E AA, (3) implies ‘fO ‘= (‘fO)_,,s+,,+t,. Now z is a ‘((almost-

period) of ‘fn therefore, according to the choice of o’ and r’ we have;

(‘fQ)Z ‘= ‘fO ‘= (‘f,)_O”+O,+z, = (3f0’,)_,‘,+rr’+s’ 3, fg,+,, 2, fO+r!

thus ‘(‘fO)T = If,,,. q

Proposition 4. If f is almost-automorphic and standard, then A,(f) is a group.

Proof. (a) If t is a ‘(almost-period) we have ‘fZ = f, moreover ‘(‘f%)-, = f by the

definition of almost-automorphy hence 1f_z = f, - z E A ,( f ). (b) If t and (T are ‘((almost-periods), ‘fO = f being almost automorphic we have

by Proposition 3: f = ‘fT = ‘(‘fO)T = ‘fg+, hence u + t E A,(f). The conjunction of (a) and (b) means that A,(f) is a group. 0

Proposition 5. If f is almost-automorphic and standard, then A,(f) is relatively dense.

Now we need Lemma 1 and 2 below. For any F c G, F finite, and any E > 0,

we denote

A(f, F, c)={t~G: WEFJf(t+t)-f(t)]<&}.

Lemma 1. Let B > 0 and F c G such that E ‘= 0 and F contains all the standard elements of G. If f is standard, then the next conditions are equivalent:

(i) A(f, F, E) is relatively dense in G, (ii) A,(f) is relatively dense in G,

(iii) VE > 0 Vfi” F c G [A(f, F, E) is relatively dense in G].

Proof. It arises immediately from the inclusions A(f, F, E) CA’(~) cA(f, F, E) with F and E standard, and from (T). 0

Lemma 2. If f is standard, then we have:

A,(f) is relatively dense in G e Vs eG12t EGS -t CA’(~).

Proof. Suppose that A,(f) 1s relatively dense in G. Let E and F be as in Lemma

1, with E and F 2ideal. The assumption that A(f, F, E) is relatively dense

becomes, by transfer:

3’*““F c G Vs E G 3t E F [(s - t) E A(f, F, E)].

By idealization the former is equivalent to:

Vs E G 12t E F [(s - t) E A(f, F, E)].

Now, A(f, I?, 2) CA,(~), the proof is ended.

Infiniresimal approach of almost-automorphic functions 293

Let us prove the converse, now. For any finite F and E > 0 standard we have:

[V.SEGG*~EG~-~EA~(~)] + Vs’sGG2t~Gs-t~A(f, F, E).

Applying idealization, we get that A(f, F, ) E IS relatively dense for any finite F, and E > 0 standard. The conclusion arises from Lemma 1. Cl

Remark. We have proved at the same time that for any 2ideal set AC1 the

following equivalence holds:

A,, is relatively dense in G e V.s E G E12t E Gs - t EA(,.

From now on, 4, and E(, are an 2ideal finite set which contains all the standard

elements of G, and an infinitesimal 2ideal strictly positive number, respectively.

We let A(, = A(f, &, Q).

Proof of Proposition 5. For any s E G, f is almost-automorphic implies ‘(“fs)_, =

5 The property “(‘J.)_.~ =f implies:

3 vu E 6) K2f)-r(u) -f(u)1 < E0 (take s = t).

By (T) we obtain:

s2t vu E 8, K2f)-t(u) -f(u)1 < Eo.

Now, vu E E, K2f,)-l(u) -f(u)1 < E,, implies ‘(“fJ)_, =f, hence we have *f,_t =

‘(‘L.-J = ‘(y.)_[ =J Th e conclusion arises from Lemma 2. 0

Theorem 3 (Geometrical characterization of almost-automorphy). Zf f is

bounded and standard, then f is almost-automorphic if and only if A,(f) is a relatively dense group.

Proof. The ‘only if’ part arises from Propositions 4 and 5. Concerning the

converse, we shall make use of condition (e) of Proposition 2:

VZ E G [3a E G 3’~’ E G (‘fO = “fxc = ‘fT and lf,_r, = f)].

Let z E G. Then we have: E13r’ E G (‘frf = If=). Let z’ satisfy the last formula; we

have to find a good cr. Since we have A,, relatively dense and 2id(A,,), according to

the remark following Lemma 2, there is a 2ideal t” E G such that z’ - t” E Ao. Let

A;, = {t”} + AC1 and 5% = {fO: o E A,‘,}. We have “fz, E cl 4, and Z$ “ideal, hence

there is a o E A,‘, such that 3f0 = “fzP. We readily verify that this o suits; indeed we

have:

(J - Z’ = (a - r”) - (z’ - .“), (a - z”) E A(,,

(t’- T”)EA~~ and A(,-A(,cA(f). 0

294 Y. Pkraire

3. What additional properties does an almost-automorphic function need from

an infinitesimal point of view, to become an almost-periodic one?

We are going to prove that an almost-automorphic function is almost-periodic if and only if the shadows of the translates are also almost-automorphic. This equivalence is similar to the equivalence of Theorem 3.3.1 in [5]. We need the following lemma.

Lemma. The external set E = nfeAA,f standard A,(f) contains a ‘ideal relatively dense in G subset, EO.

Proof. Let F be a finite ‘ideal set containing all the standard elements of G. Let 9 be a ‘ideal finite set of almost-automorphic functions which contains all the standard automorphic functions. For any t E F, f E 9 we have ]ff -f) E AA, hence:

4s) = yayEx MS) -.W)I E AA. l

The function v(s) is ‘ideal and v(0) = 0. Let EO = A(v, G, E) where G is a finite set which contains the 2ideal elements of G, and let E 2= 0, E > 0. We have:

Eo = AZ(V) ~~~~~~~~~~~~~~ Al(f 1.

Clearly v E AA, consequently we have:

%I &c (

n A,(f) & EO relatively dense in G >

. f EAA, f standard

Making use of the principle of transfer we obtain:

feAA,fstandardAl(f) & EO relatively dense in G f-l >

. 0

Theorem 4. If f is standard, then f is almost-periodic if and only if, for any s ‘fs is

almost-automorphic.

Proof. The necessity arises from the fact that, if f is almost-periodic, then each ‘fs = Ifs is almost-periodic and in particular almost-automorphic. Suppose that each rh is almost-automorphic. The lemma give us a relatively dense *ideal set EC,

such that EO c n A,(%). For any s E G, t E EO we have, by Proposition 3 in Section 2:

(%)(t) = (Y&(O) %+1(O) =f (s + t).

Hence, t being an ‘(almost-period) of IL, we have:

f(s) = (%)(O) = (X)(t) =f (s + t).

Hence, the relatively dense set E,, is included in P,(f) and f is almost- periodic. Cl

Znfinitesimnl approach of almost-automorphic functions 295

Here is an example of an almost-automorphic but not almost-periodic function. This example given without proof in [5, p. 7201 is due to Furstemberg.

Let f: Z- R be the function defined by f(n) = sgn(sin(n)). We have

f(n) = 1 if sin(n) > 0,

f(n) = -1 if sin(n) < 0,

f(0) =o if sin(n) = 0.

The equality sin(n) = 0 occurs only for n = 0. Let us give a non-standard proof that f is almost-automorphic but not

almost-periodic. (a) f is almost-automorphic. The function f is standard; the equality

‘(‘fm)-,=f. IS o vrous for standard m, so it suffices to prove it for non-standard b .

m. So assume ]m] is infinitely large.

Vln ‘[fm(n)] = ‘sgn(sin(n + m)) = sgn(sin(n + m)) = f (n + m).

Let cx be a real standard number such that sin(a) = ‘[sin(m)]; such an a: exists because sin(m) belongs to the compact interval [ -1, 11. So we have cos( a) = ‘[cos(m)] and, using the classic trigonometric identities:

V’n E Z sin(n + m) ‘= sin(n + (u). (1)

From (1) we infer:

‘fm(n) = ‘[f,(n)] = sgn(sin(n + m)) = sgn(sin(n + a)), (2)

in any case where sin(n + CX) # 0. The case sin(n + (Y) = 0 is exceptional; it happens, since x is irrational, for one value n, of the variable at most. The functions ‘fm and sgn(sin(. + (w)) being standard we get from (2) by transfer, that for any n such that it # no:

‘fm(n) = sgn(sin(n + (u)).

Thus, in particular:

(lfm)-m(n) = sgn(sin(n - m + a)).

Let us prove that:

(3)

V’n E Z sin(n - m + CX) ‘= sin(n).

Let n be standard; then

(4)

sin(n - m + a) = sin(n - m) cos(a) + cos(n - m) sin(a)

= [sin(n) cos(m) - sin(m) cos(n)] cos( a) + [cos(n) cos(m) - sin(m) sin(n)] sin( a)

‘= [sin(n) cos( cu) - sin(a) cos(n)] cos( (u) + [cos(n) cos( (r) + sin(a) sin(n)] sin(a)

= sin(n) (sin”(a) - cos2(cr)) = sin(n).

296 Y. Pe’raire

As Irnl is infinitely large, we have II -m # n,,. Hence sin(n - m + a) #O. According to (3) and (4) it follows:

V’n E Z (Ifm)_m(n) = sgn(sin(n -m + a)) = sgn(sin(n)) =f(n)

which can also be written, 1(1fm)_m =& So S is almost-automorphic.

(b) f is not almost-periodic. The integers being densely distributed modulo 2x,

there exists an integer m, m # 0, such that sin(m) ‘= 0 and cos(m) ‘= 1.

Consequently we have sin(n + m) ‘= sin(n) for any standard II. Now,

V’n # 0 sgn(sin(n + m)) = sgn(sin(n)), because sin(n) # 0.

Therefore, by transfer, if E = sgn(sin(m)), then for any n, standard or not:

(‘f,)(n) =f(n) if n f 0,

(%)(O) = & = fl.

This implies that, if (I&) exists, then (tfm) # (‘fm) because (l&)(-m) ’ =

f,(-m) =f(O) = 0 and (I&)(-m) =f(-m) # 0. Hence f is not almost-periodic.

4. Equivalence of our definition of almost-automorphy with the classic

definitions

We shall prove that the geometrical characterization given by Theorem 3, is

equivalent to the Bohr almost-automorphy.

(a) Let f be a standard function such that A,(f) is a relatively dense group;

then:

(i) for any subset F of G and any E >O, we deduce from Lemma 1 that

A(& F, E) is relatively dense in G,

(ii) for any standard finite subset F of G and any standard real number E > 0,

we have

A,,-A,cA,(f)-A,(f)cA,(f)cA(f,F, E)

because A,(f) is a group. Hence, by (T), we get that for any finite subset F of G

and any E > 0, there is a finite superset F’ of F and E’ > 0 such that

A(f, F’, E’) - A(f, F’, E’) c A(f, F, E).

Hence f is Bohr almost-automorphic.

(b) Conversely, if the standard function f is Bohr almost-automorphic, it arises

from Lemma 1 that A,(f) is relatively dense in G. Moreover if s, t E A,(f) we have s, t E A(f, F, E) for any standard finite F and any standard real number

E > 0; according to condition (ii) of the definition of Bohr almost-automorphy,

s - t E A(f, F, E) holds for any standard finite F and for any standard real number

E > 0. The last condition is equivalent to s - t E A,(f). This completes the proof.

Infinitesimal approach of almost-automorphic functions 297

References

[I] S. Bochner, A new approach of almost periodicity, Proc. Nat. Acad. Sci. USA 48 (1962).

[2] Y. PCraire, ThCorie relative des ensembles internes, Osaka J. Math. 29 (2) (1992).

[3] I’. PCraire, Polytransfert et principe de comprChension dans la thCorie relative des ensembles

internes, sumitted to J. Symbolic Logic.

(41 Y. PCraire, Analyse relative, Annales scientifiques de I’Universi6 de Clermont-Ferrand 98 (28)

(1992). [5] W.A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965) 719-751.