The law of the iterated logarithm for c k f(n k x)

The law of the iterated logarithm for k fn k x) Christoph Aistleitner Abstrat By a lassial heuristis, systems of the form osπn k x) k 1 and fn k x)) k 1, where n k ) k 1 is a rapidly growing sequene of integers, show probabilisti properties similar to those of independent and identially distributed i.i.d.) random variables. For example, Erdős and Gál proved the law of the iterated logarithm LIL) in the form limsup N log log N) 1/ N osπn kx = 1/ a.e., valid for n k ) k 1 satisfying the launary growth ondition n k+1 /n k > q > 1, k 1. Weiss extended this to limsup BN log log B N) 1/ N osπn kx = 1 a.e., again for launary n k ) k 1, where BN = N k, under the additional assumption N = ob N /log log B N ) 1/ ) as N. This diretly orresponds to a general LIL for i.i.d. random variables due to Kolmogoroff. In this paper we generalize Weiss s result to systems fn k x)) k 1, where f is a funtion of bounded variation, under an almost best possible growth ondition for the oeffiients k ) k 1, thus partially solving a problem posed by Walter Philipp in his famous paper of 1975. 1 Introdution A inreasing sequene n k ) of positive integers is alled a launary sequene if it satisfies the Hadamard gap ondition n k+1 n k > q > 1, k 1. A lassial heuristis states that the systems os πn k x) k 1 or fn k x)) k 1, 1) where n k ) k 1 is a launary sequene of integers and f is a nie funtion, exhibit properties similar to those of systems of independent and identially distributed i.i.d.) random variables. For example, Erdős and Gál [6] proved in 1955 that for a launary sequene n k ) k 1 N os πn kx N log log N = 1 a.e., Graz University of Tehnology, Institute of Mathematis A, Steyrergasse 3, 81 Graz, Austria. e-mail: aistleitner@finanz.math.tugraz.at. Researh supported by the Austrian Researh Foundation FWF), Projet S963-N13. This paper was written during a stay at the Alfréd Rényi Institute of Mathematis of the Hungarian Aademy of Sienes, whih was made possible by an MOEL sholarship of the Österreihishe Forshungsgemeinshaft ÖFG). Mathematis Subjet Classifiation: Primary 11K38, A55, 6F15 Keywords: launary series, law of the iterated logarithm 1

whih is similar to the law of the iterated logarithm for i.i.d. random variables, stating that for an i.i.d. sequene X 1,X,... satisfying EX 1 =, EX 1 = σ <, N X k N log log N = σ a.s. ) If the funtion os πx is replaed by another funtion fx) satisfying fx + 1) = fx), fx) dx =, and if f is additionally Lipshitz ontinuous Takahashi [1], 196) or of bounded variation on [,1] Philipp [1], 1975), then N fn kx) N log log N C a.e. 3) for some onstant C. On the other hand, there are examples showing that an exat law of the iterated logarithm LIL) like ) will not neessarily hold in the ase of general funtions fx) instead of os πx: hoose e.g. fx) = os πx + os πx, and n k = k + 1, k 1. Then as was pointed out by Erdős and Fortet. N fn kx) N log log N = os πx a.e., ) There exists an important generalization of the LIL ) to the ase of independent, not identially distributed random variables, proved by Kolmogoroff [7] in 199. Let X 1,X,... be independent random variables satisfying EX k =, EXk = σ k < k 1), and define N 1/ B N = σk), N 1. Then N X k = 1 a.s., 5) BN log log B N provided B N and there exists a sequene m k suh that ) B N X k m k, k 1, and m N = o log log BN as N. There are several possibilities to modify 5) to our situation of systems of the form 1). One way is to bound the number of elements n k lying in dyadi intervals of the form [ r, r+1 ), r = 1,,... The author proved, together with I. Berkes [], the following result: Let fx) be a funtion satisfying fx + 1) = fx), fx) dx =, Var [,1] f. 6)

Define and Then a N,r = #k N : n k [ r, r+1 )}, r, N 1, 1/ B N = an,r), N 1. r= N fn kx) B N log log B N C a.e. for some onstant C, provided for some onstant α > 3, uniformly for r N. a N,r = O B N log N) α) Another possibility is to introdue oeffiients k ) k 1 and onsider k os πn k x) k 1 or k fn k x)) k 1 instead of 1). In 1959 Weiss [13] proved the following: Let B N = 1 and assume B N as N. Then 1/ N k), N 1, N k os πn k x B N log log B N = 1 a.e., 7) provided ) B N k = O log log BN as N, in perfet analogy with Kolmogoroff s result the upper bound in 7) was already shown in 195 by Salem and Zygmund [11]). It is reasonable to assume that a result similar to 7) should also hold if the funtion os πx is replaed by fx) for f satisfying 6). In his famous paper [1] from 1975 entitled Limit theorems for launary series and uniform distribution mod 1 Walter Philipp stated the problem in the following form: Give a detailed proof that N kfn k x) B N log log B N 1 a.e., 8) 3

where B N = N N 1/ k), ) B N = o. 9) log log BN The purpose of this paper is to give a partial solution of the problem, and to verify 8) under a ondition slightly stronger than 9): Theorem 1 Let fx) be a funtion satisfying 6), and let n k ) k 1 be a launary sequene of integers. Then N kfn k x) C a.e., BN log log B N for some onstant C, provided and N 1/ B N = k) 1) N = O B N log log B N ) 3/ ). 11) In fat, we are not sure if the theorem would really remain true with 11) replaed by 9). The problem to find the best possible upper bound for N in 11) remains unsolved. In view of 3) and ) it is lear that no stronger result than 8), i.e. no exat LIL like in ) and 7) an be expeted in our ase. Nevertheless, we know that the exat law of the iterated logarithm for fn k x)) k 1 for launary n k ) k 1 and f satisfying 6) is valid in the form N fn kx) N log log N = f a.e., provided the number of solutions of Diophantine equations of the type an k ± bn l =, a,b, Z, k,l N, is not too large ompared with N []; f. also [1], [3]). It is reasonable to assume that under a similar number-theoreti ondition it is possible to prove an exat law of the iterated logarithm also for systems of the form k fn k x)) k 1. Preliminaries Without loss of generality we assume that f is an even funtion, i.e. the Fourier series of f an be written in the form fx) a j os πjx j=1

the proof in the general ase is exatly the same). Sine by assumption Var [,1] f the Fourier oeffiients of f satisfy a j j 1, j 1 1) f. Zygmund [1, p. 8]). We write px) for the J-th partial sum of the Fourier series of f, and rx) for the remainder term, i.e. px) = J a j os πjx, rx) = j=1 j=j+1 a j os πjx. The value of J will be determined later. Throughout this setion we will assume that N is fixed. For the funtion p we have p f + Var [,1] f 3, 13) by.1) of Chapter II and 1.5) and 3.5) of Chapter III of Zygmund [1], independent of J. Lemma 1 1MN M k rn k x) B NJ 1/. here and in the sequel, the onstant implied by the symbol must not depend on N,J, but may depend on q,f). Proof: By the orthogonality of the trigonometri system, 1), Minkowski s inequality and the Carleson-Hunt inequality see e.g. Mozzohi [9] or Arias de Reyna [5]) we have M 1MN k rn k x) i= M 1MN k a j osπjn k x) j [Jq i,jq i+1 ) M k a j osπjn k x) i= j [Jq i,jq i+1 ) M k j 1 osπjn k x) i= B N J i= B N J 1/. q i/ j [Jq i,jq i+1 ) Observe that the Carleson-Hunt inequality allows us to eliminate the in 1). This is possible beause splitting the Fourier series of rx) into parts ontaining only frequenies in one interval of the form [Jq i,jq i+1 ) for some i guarantees that for k 1 > k always 1) 5

j 1 n k1 Jq i n k1 > Jq i qn k Jq i+1 n k, provided j 1,j [Jq i,jq i+1 ) for some i. Now we hoose As a onsequene of Lemma 1 we easily get J = JN) = log B N 6. 15) Lemma P x,1) : 1MN } M rn k x) > B N J 1 log B N ) 6. Here and in the sequel, P denotes the Lebesgue-measure on,1). 3 Exponential inequality We still assume that N is fixed. By 11) there exists a onstant C 1 suh that N C 1 B N log log B N ) 3/, N 1. 16) By the hoie of J in 15) it is possible to find a small number C, whih must not depend on N,J, suh that C log log BN log log B N) 3/ 1 B N 6C 1 B N log q J). 17) We write ex) = e x. Lemma 3 ) N e λ k pn k x) dx e λ CBN), for some number C independent of J,N), provided λ C log log BN. 18) B N Proof: Divide the integers 1,,...,N into bloks 1,,..., w for some appropriate w), suh that every blok ontains log q J) numbers the last blok may ontain less), i.e. 1 = 1,..., log q J) }, = log q J) + 1,..., log q J) },... Write I 1 = I = e λ e λ i odd k pn k x) dx k pn k x) dx. 6

Then by the Cauhy-Shwarz-inequality ) 1 N e λ k pn k x) dx I 1 I ) 1/. 19) Writing we have I 1 = = where we used the inequality U i = k pn k x), i 1, e λ k pn k x) dx 1 + λ k pn k x) + λ k pn k x) dx 1 + λui + λ Ui ) dx, ) e x 1 + x + x, valid for x 1, and the fat that 11), 13), 17) and 18) imply λu i λ k p ) 6λ i k C 1 B N 6λ log q J) log log B N ) 3/ 1 i denotes the number of elements in i ). Now Ui = J a j os πjn k x = k j=1 k 1,k i 1j 1,j J k1 k a j1 a j ) os πj1 n k1 + j n k )x + os πj 1 n k1 j n k )x 1) = V i + W i, ) say, where V i is a sum of trigonometri funtions having frequenies in [n i,jn+ i ], and W i is a sum of trigonometri funtions having frequenies in [,n i ) here n i denotes the smallest 7

and n + i the largest number in i ). No other frequenies an our, sine the largest possible frequeny in 1) is Jn + i + Jn + i = Jn + i. We note that the frequenies of the trigonometri funtions in U i are also in the interval [n i,jn+ i ], and write X i = λu i + V i 3) Using Minkowski s inequality we have W i 1 k 1,k i 1j 1,j J }} j 1 n k1 j n k <n i k 1,k i,k 1 k 1j 1,j J } } j 1 >j n k /n k1 1 k 1,k i,k 1 k π 6 q v= J j=1 1 q v k1 k a j1 a j k1 k 1 j 1 j k1 k n k1 j n k. ) Let i 1 < i be two distint even numbers. Then the frequeny of any trigonometri funtion in X i is at least twie as large as the frequeny of any trigonometri funtion in X i1. In fat, the largest trigonometri funtion in W i1 is at most Jn + i 1, and the smallest trigonometri funtion in X i at least n i, and sine we have mink i } k i1 } log q J) n i > q log q J) n + i 1 Jn + i 1. This implies that for any distint i 1,...,...,i v v is arbitrary), all even, the funtions X i1,...,x iv are multipliatively orthogonal, i.e. From ), ), 3), ) and 5) we onlude I 1 X i1 X iv dx =. 5) 1 + Xi + λ W i ) dx 1 + X i + 16λ q 8 dx

= e 1 + 16λ q e 16λ q 16λ q dx dx A similar estimate for I an be obtained in the same way, and finally 19) yields ) 1 N e λ k pn k x) dx e = e = e 16λ q 8λ q 8λ ) q B N, e i odd 16λ q 1/ whih proves the lemma. Lemma There exists a large number C 3, independent of J,N, suh that } N P pn k x) > C 3 BN log log B N log B N ) 6 Proof: In Lemma 3 we hoose λ = C log log BN, B N whih is onsistent with 18), and get N } P pn k x) > C 3 BN log log B N = P e λ ) N pn k x) ) > e λc } 3 BN log log B N 8λ ) q e B N λc 3 BN log log B N ) 8qC = e log log B N C C 3 log log B N 9

e 6log log B N ) = log B N ) 6 6) for suffiiently large C 3. A results similar to Lemma 3 is possible for N pn kx) instead of N pn kx), whih yields } N P pn k x) > C 3 BN log log B N log B N ) 6. 7) Combining 6) and 7) we get the lemma. Proof of Theorem 1 We define a sequene N 1,N,... reursively in the following way: Let N 1 = 1, and for m 1 let Nm + 1 if N N m+1 = m+1 m1/3 ) M > N m : M )} k=n m+1 k < m1/3 otherwise This means that always N m+1 1 k=n m+1 < m1/3 ) and N m+1 k=n m+1 m1/3 ) 8) the sum on the left side may be over an empty index set), and in partiular B N m m v1/3) m1/3) m /3. 9) v=1 Also, 16) guarantees, together with 8), that whih in partiular implies and B N m+1 B N m + m1/3) + B N m m1/3) m /3 C 1 B Nm+1 log log B Nm+1 ) 3/ B Nm+1 B Nm 1 as m. 3) For C > C 3 3 we apply the results from the previous two setions for N = N m+1 ) and get } M P k fn k x) > C BN m log log BN m 31) N m+1mn m+1 1

} M P N m+1mn m+1 k pn k x) > C 1) BN m log log BN m } M +P N m+1mn m+1 k rn k x) > B N m } N m P k pn k x) > C 3) BN m log log BN m M +P N m+1mn m+1 1 k pn k x) k=n m+1 > BN m log log BN m +P Nm+1 pn kx) } > B log log B Nm Nm +log B Nm ) 6 P N m+1mn m+1 1 +log B Nm ) 6. M k=n m+1 k pn k x) > BN m log log BN m 3) 33) 3) 35) Here we used Lemma to estimate 3), Lemma to estimate 33), and 3) vanishes for suffiiently large m beause of 11) and 3). It remains to find an appropriate estimate for 35). We have = M N m+1mn m+1 1 k pn k x) k=n m+1 log B M Nm+1 6 N m+1mn m+1 1 k a j osπjn k x) k=n m+1 j=1 log B Nm+1 6 M a j j=1 N m+1mn m+1 1 k osπjn k x) k=n m+1 log B Nm+1 6 1 M j j=1 N m+1mn m+1 1 k osπn k x) k=n m+1 M log log B Nm+1 N m+1mn m+1 1 k osπn k x) k=n m+1 log q 1 log log B Nm+1 N m+1mn m+1 1 k osπn k x) s= N m+1km k s mod log q 11

log q 1 log log B Nm+1 log log B Nm+1 s= log q 1 s= N m+1mn m+1 1 N m+1kn m+1 1 k s mod log q k osπn k x) N m+1km k s mod log q k osπn k x), 36) where the last estimate follows from the Carleson-Hunt inequality. If two distint integers k 1,k are in the same residue lass mod log q ) then neessarily n k1 /n k [1/,]. Thus N m+1kn m+1 1 k s mod log q N m+1k 1,k,k 3,k N m+1 1 k 1,k,k 3,k s mod log q N m+1k 1,k,k 3,k N m+1 1 k 1,k,k 3,k s mod log q N m+1k 1,k N m+1 1 k 1,k s mod log q N m+1kn m+1 1 k s mod log q N m+1kn m+1 1 m1/3 ) ) B Nm m /3 ) k osπn k x) 1 dx k1 k k3 k 1n k1 ± n k ± n k3 ± n k = ) k1 k k3 k 1n k1 n k = ) 1n k3 n k = ) by the definition of N m,n m+1, whih implies, in view of 3) and 36), N m+1mn m+1 1 M k=n m+1 k pn k x) B N m log log B Nm m 1/3. 1

Thus P N m+1mn m+1 1 M k=n m+1 k pn k x) > BN m log log BN m log log B N m ) m 1/3 ) log m) m /3. 37) Writing A m for the set in 31), m 1, i.e. A m = x,1) : by 35) and 37) we have and thus N m+1mn m+1 } M k fn k x) > C BN m log log BN m, PA m ) log B Nm ) 6 + log m) m /3 PA m ) <. m=1 m 1/3) 6 + log m) m /3 Therefore the Borel-Cantelli-lemma implies that M N m+1mn m+1 1 k fn k x) > C BN m log log BN m for at most finitely many m for all x,1), exept a set of measure zero. Thus, in view of 3), if C 5 > C we also have N k fn k x) > C 5 BN log log B N for only finitely many values of N, again for all x,1) exept a set of measure zero. Therefore we have N kfn k x) 1 a.e., C 5 BN log log B N whih proves Theorem 1. 13

Referenes [1] C. Aistleitner. Irregular disrepany behaviour of launary series. Monatshefte Math., to appear. [] C. Aistleitner. On the law of the iterated logarithm for the disrepany of launary sequenes. Trans. Amer. Math. So., to appear. [3] C. Aistleitner and I. Berkes. On the entral limit theorem for fn k x). Probab. Theory Related Fields, 16: 67-89, 1. [] C. Aistleitner and I. Berkes. On the law of the iterated logarithm for the disrepany of n k x. Monatshefte Math. 156, no. : 13-11,9. [5] Arias de Reyna. Pointwise onvergene of Fourier series. Leture Notes in Mathematis, 1785. Springer-Verlag, Berlin,. [6] P. Erdős and I.S. Gál. On the law of the iterated logarithm. I, II. Nederl. Akad. Wetensh. Pro. Ser. A. 58 = Indag. Math. 17: 65 76, 77 8., 1955. [7] A. Kolmogoroff. Über das Gesetz des iterierten Logarithmus. Math. Ann. 11:16-135, 199. [8] F.A. Móriz, R.J. Serfling and W.F. Stout. Moment and probability bounds with quasisuperadditive struture for the imum partial sum. Ann. Probab. 1,no. :13-1, 198. [9] C.J. Mozzohi. On the pointwise onvergene of Fourier series. Leture Notes in Mathematis, Vol. 199. Springer-Verlag, Berlin-New York, 1971. [1] W. Philipp. Limit theorems for launary series and uniform distribution mod 1. Ata Arith. 6:1-51, 1975. [11] R. Salem and A. Zygmund. La loi du logarithme itéré pour les séries trigonométriques launaires. Bull. Si. Math. 7, 9, 195. [1] S. Takahashi. An asymptoti property of a gap sequene. Pro. Japan Aad. 38:11-1, 196. [13] M. Weiss. The law of the iterated logarithm for launary trigonometri series. Trans. Amer. Math. So. 91: 69, 1959. [1] A. Zygmund, Trigonometri Series. Vol. I, II. Reprint of the 1979 edition. Cambridge Mathematial Library. Cambridge University Press, Cambridge, 1988. 1