MEASURE AND CATEGORY { Department of Mathematics, University of Tennessee, Chattanooga,

Size: px

Start display at page:

Download "MEASURE AND CATEGORY { Department of Mathematics, University of Tennessee, Chattanooga,"

Erik Wilkerson
6 years ago
Views:

1 Mathematica Pannonica 7/1 (1996), 69 { 78 MEASURE AND CATEGORY { SOME NON-ANALOGUES Harry I. Miller Department of Mathematics, University of Tennessee, Chattanooga, TN, , USA Franz J. Schnitzer Institut fur Mathematik, Montanuniversitat, 8700 Leoben, Austria Received March 1995 MSC 1991: 40 A 05, 40 C 05 Keywords: Summability, non-schur matrices, sets of rst category. Abstract: Several results from the area of innite series are established showing that somesubsets ofthe interval [0 1] for which certain series are convergent are of rst category. 1. Introduction \Measure and Category" is the title of a well{known and highly interesting book by J.C. Oxtoby [8]. In it onends numerous analogues and non-analogues between measure and category. A non-analogue that seems not widely known and does not appear in Oxtoby's book is the following. For each x 2 (0 1] let x = e n (x) 2 n be the non-terminating binary representation of x. The number x is said to be normal to base 2, in case the following limit exists:

2 70 H. I. Miller and F. J. Schnitzer 1 lim n!1 n nx k=1 e k (x) = 1 2 : It is known for quite some time that the Lebesgue measure of the set N 2 = x 2 [0 1] x : normal to base 2 is equal to 1. Onthe other hand, it seems less well known that N 2 is a set of rst Baire category. Hence, N 2 is large in measure but small in category in this case there is a non-analogue between measure and category. Another result concerning series is the following: It is long known ([4]) that if converges then the random series a 2 n a n converges for almost all choices of the signs + and ;. More precisely: If e n (x) x = x 2 (0 1] 2n is again the non-terminating binary expansion of x, then the set M = ( has measure one. x 2 (0 1] 1 X (2e n (x) ; 1)a n is convergent ) 2. Results Our rst result shows that the set M just dened might be of the rst Baire category since the following is true: Theorem 1. If a n is not absolutely convergent, then the set

3 A = ( x 2 [0 1] 1 X Measure and category 71 (2e n (x) ; 1)a n is convergent is of the rst category. Proof. The seta can be expressed in the following form: A= 1\ k=1 1[ ( x 2 [0 1] jx (2e n (x) ; 1)a n 1 < = n=i 1\ k=1 1[ A N k! ) k 8j >i N )! the denition of the sets A N k being obvious. It will be shown that each A N k is nowhere dense. Take any Since the series `extension' of x such that 1P holds. Denoting by x = a n sx n=m+1 mx x n 2 n where m>n: is not absolutely convergent, there exists an x 0 = x + (2y n ; 1)a n = n=m+1 y n 2 n sx n=m+1 ja n j 1 k B(s) = z 2 [0 1] en (z) =x n for n =1 2 ::: m and e n (z) =y n for n = m +1 m+2 ::: s it follows that B(s) \ A N k =. Since the choice of x was arbitrary and since the setb(s) isaninterval, the sets A N k are nowhere dense and hence A is a set of rst category. If S = fs k g is a sequence and ifthe arbitrarily chosen x 2 (0 1] has the non-terminating binary representation e n (x) x = 2 n then the subsequence of S determined by x and denoted by S(x) is dened to befs kj g 1 j=1 where k j is the place where the j-th oneappears =

4 72 H. I. Miller and F. J. Schnitzer in the sequence fe 1 (x) e 2 (x) :::g. Buck andpollard [2], have proved in 1943 the following Theorem A. If S = fs k g is a (C 1)-summable sequence for which k=1 S 2 j k k 2 < 1 holds, then the Lebesgue measure of the set x 2 [0 1] S(x)is (C 1)-summable to the (C 1)-limit of S is one. Szusz [10] later, in 1968, gave a simple probabilistic proof of this result using the strong law of large numbers of Kolomogorov. Recently, the current authors rediscovered the fact that the category analogue of the Th. A does not hold. Namely, ifs = fs k g is a divergent sequence and C is a regular matrix summability method then the set A = x 2 (0 1] S(x) isc-summable is of the rst category. This result was rst published in [6]. We are indebted to Professor J. Fridy for pointing outthis reference to us. He also informed us about a paper by T. Keagy, [5], containing the following two results. Theorem K 1. If A is a non-schur matrix with convergent column and if S = fs k g is a divergent sequence, then the set fx 2 [0 1] S(x) is A-summableg is of the rst category. Remark. Since the class of non-schur matrices with convergent columns strictly contains the class of regular matrices, Th. K 1 is a generalization of the theorem mentioned above that appears in [6]. Theorem K 2. Let A be a non-schur matrix with convergent columns and let S = fs k g denote a divergent sequence. Then the set of A- summable rearrangements of S is of the rst category. Remark. Here a word about terminology seems in order. Let P denote the collection of all permutations of N, the set of natural numbers, i.e. the collection of all injective mappings P of N onto itself. On P a metric d is dened in the following way: if P 1 P 2 2 P then d(p 1 P 2 )=0 in case P 1 = P 2 whilst d(p 1 P 2 )= 1 if P 1 (i) =P 2 (i) j if i =1 2 ::: j; 1 and P 1 (j) 6= P 2 (j). In the metric space (P d) P is then of the second category in itself [3].

5 Measure and category 73 Our next results Th. 2, 2 0 and 3 areextensions of Th. K 1. Every sequence S = fs k g is summed by some non-schur matrix with convergent columns then the following holds: x 2 (0 1] S(x) isa-summable for some non-schur matrix with convergent columns equals the interval (0 1]. Here it will be shown that ifs = fs k g is a divergent sequence, then there exists a collection A of non-schur matrices with convergent columns, having cardinality of the continuum, such that the set x 2 (0 1] S(x) isa-summable for some A 2 A is of rst category. Of course, this result would be trivial in case all of the matrices in A had the same convergence eld. In connection with this remark the following example is informative. Example 1. Let ">0. For t 2 [0 ")dene A t = a nk (t) asfollows: a nk (t) = 8 >< >: 1+(;1) n+1 (t ; ") for k =1 2 ::: n (;1) n t m 1 (n) for k 2 B 1 (n) (;1) n+1 " for k 2 B 2 (n) m 2 (n) 0 otherwise, where B 1 (n) and B 2 (n) are for each n blocks of consecutive integers satisfying: a) B 1 (n) liestothe `left' of B 2 (n) b) n is less than each integer in B 1 (n) c) all of the blocks fb 1 (n) n 2 Ng and fb2 (n) n 2 Ng are pairwise disjoint d) m i (n) isthe number of elements ofb i (n), i =1 2 S e) the set 1 (B 1 (n) [ B 2 (n)) has natural density 0inN f) lim n!1 m i(n) =1 for i =1 2. A t is a regular matrix summability method for each t 2 [0 "). For each such t let x t denote a sequence of 0's and 1's satisfying

6 74 H. I. Miller and F. J. Schnitzer 1 m 1 (n) 1 m 2 (n) X k2b 1 (n) X k2b 2 (n) x t (k) ;! " as n!1 and x t (k) ;! t as n!1: From this it is easy to derive that: x t is A t -summable to 0 for t 2 [0 "), but x t is not A s -summable if 0 t, s<"and t 6= s hold. In the sequel the following notations will be used: K(A ") := B B is a non-schur matrix with convergent where jjcjj = sup p columns and jja ; Bjj <" jc pq j! if C =(c pq ): Now our extensions of Th. K 1 will be presented. Theorem 2. If A is a non-schur matrix with convergent columns and if S = fs k g is a bounded divergent sequence, then the set x 2 (0 1] S(x)is B-summable for some B 2 K(A ") is of the rst category for some ">0. Proof. This proof follows the lines of the proof of Th. 3 in [5] and is divided into the consideration of two cases. Let A =(A pq )and let T (A ") denote the set x 2 (0 1) S(x) isb-summable for some B 2 K(A "). Further, J shall denote the collection of all subsequences of S. CASE I. Supposerow p of A is not absolutely convergent. If y 2 J and y = fy q g is B-summable for some B 2 K(A "), B =(B pq ), then there exists ann such that jx b pq y q 1 for every j>in: 2 jx a pq y q 1 for every j>in: From this follows that if">0 is suciently small, then Next, the following set is dened:

7 E N = Measure and category 75 nx 2 (0 1] there exist j>insuch that jx a pq (S(x)) q > 1 o: By the above T (A ") 1[ E c N : Moreover, in [5] it is shown that the set on the right sideisofthe rst category, sothat the samemust hold for T (A "). CASE II. Supposeeach rowofa to beabsolutely convergent. Now Maddox (Lemma 1 in [5]) showed that ifthe matrix A is Schur there exists a divergent sequence x such that eachsubsequence of x is A- summable. Hence there is an y 2 J such that y is not A-summable. Therefore, there exists a >0suchthat toevery positive integer N integers j N >i N N exist so that () holds. Next, let be E N = ( a in qy q ; a jn qy q > x 2 (0 1] there exists j>in such that ) a iq (S(x)) q ; a jq (S(x)) q > : 2 Since each rowofa is absolutely convergent andsinces is bounded it follows that each x 2 E N is contained in an interval that isasubset of E N. By () andthe fact that the columns of A converge follows that each E N is dense in (0 1). Now suppose w 2 J, w being B-summable, B =(b pg )and B 2 2 K(A "). Then there exists ann such that b iq w q ; b jq w q 4 is true for all j>i N. This implies that a iq w q ; a jq w q 2 holds for every j>i N, if only " is suciently small. Thus it has been shown that

8 76 H. I. Miller and F. J. Schnitzer T (A ") 1[ and hence T (A ") isofthe rst category. The following example shows that the condition of S being boundedinth.2isnecessary. Example 2. Let S n = n, n =1 2 ::: and suppose that y = fy q g is any subsequence of S = fs n g. By C denote the (C 1)-matrix. It will be shown that for each ">0there exists a regular matrix method C " 2 K(C ") sothat C " sums y to zero. For n>2the matrix method C(n) isdened as follows: the rst n ; 1rows of C(n) are the sameas those of C. Then there exists aq n >nso that 1 nx y q ; 1 n n y q n < 0: E c N Therefore there exists a n,0< n < 1, such that 1 nx y q ; n n n y q n =0: The terms in the n-th rowofc(n) are to be 1 n in the rst n places, ; n n in the q n -th place and 0 in all other places. The remaining rows are dened similarly yielding with C(n) =(c (n) pq ) c (n) pq y q =0 for all p n: Obviously, each C(n) is regular and jjc ; C(n)jj! 0asn!1. Despite this example, for unbounded sequences we have the following result: Theorem 2 0. If A is a non-schur matrix with convergent columns and if S = fs k g is an unbounded sequence, then for every ">0 there exists acollection A, having cardinality of the continuum, A K(A "), such that the set x 2 (0 1] S(x)is B-summable for some B 2 A is of the rst category. Proof. Let "> 0 and let p be any positive integer. There exists an A 0 =(a 0 pq), A 0 2 K(A " 2 )such that row p of A0 is not eventually zero. There also exists acollection A 2 K(A "), A having cardinality ofthe continuum, such that ifb 2 A and a 0 pn 6=0,then jb pnj > 1 2 ja0 pnj. Let next be:

9 Measure and category 77 E N = x 2 (0 1] there exists ann N such that ja 0 pn(s(x)) n j > 1. Clearly, each E N is dense in (0 1] and if x 2 E N then an interval containing x is contained in E N.Theset x 2 (0 1] S(x)is B-summable for some B 2 A 1S is obviously a subset of EN c and istherefore of the rstcategory. We nowturn our attention to theorems dealing with rearrangements of sequences. If S = fs n g and P 2 P, let S(P )denote the rearrangement ofs by P, i.e. (S(P )) k = S P (k) for each k. Theorem 3. If A is a non-schur matrix with convergent columns and if S = fs k g is a bounded divergent sequence, then the set U(A ") = P 2 P S(P ) is B-summable for some B 2 K(A ") is of the rst category in (P d) for some ">0. Proof. CASE I. Suppose row p of A is not absolutely convergent. If y =(y q ) 2 H, where H is the collection of all rearrangements ofs, y being B-summable for some B 2 K(A "), B =(b pq ), then there exists an N such that jx b pq y q 1 for every j>in: 2 jx b pq y q 1 for all j>in hold. From this follows that, for ">0 suciently small, the inequalities Now let E N = np 2 P there exist j>in such that jx >1 a pq (S(P )) q o: Notice that ifp 2 E N, P 0 2 P and P (q) =P 0 (q) for all q =1 2 ::: j, then P 0 2 E N. Hence, by the denition of the metric d on P, E is an open set. Furthermore, by the argument used by Keagy in proving Th. 4 in [5], it follows that each E N is dense in P. Therefore it has been shown that U(A ") that 1S EN c 1 S EN c and also U(A ") isofthe rst category. holds for suciently small " and

10 78 H. I. Miller and F. J. Schnitzer: Measure and category CASE II. Suppose each row of A is absolutely convergent. In this case the proof then follows exactly that in Case II in Th. 2. In place of the lemma of Maddox in [5], the following result of Keagy, [5], is used: If A is a non-schur matrix and ifs = fs k g is a divergent sequence then there exists a P 2 P such that S(P )is not A-summable. Example 2 shows also that the requirement that S is bounded in Th. 3 is necessary. However, we again do have the following Theorem 3 0. If A is a non-schur matrix with convergent columns and if S = fs k g is an unbounded sequence, then for every ">0 there exists acollection A of cardinality of the continuum, A K(K ") such that the set P 2 P S(P )is B-summable for some B 2 A is of the rst category. The proof of this result isthe same asthat ofthe proof of Th. 2, only with x replaced by P and (0 1] replaced by P. Two nal remarks 1. The rearrangement analogue of Th. 1 was proved by Agnew in [1]. 2. A recent result ofthe type of results presented here is by Miller and can found in [7]. References [1] AGNEW, R. P.: On rearrangements of a series, Bull. Amer. Math. Soc. 46 (1940), 797{799. [2] BUCK, R. C. and POLLARD, H.: Convergence and summability properties of subsequences, Bull. Amer. Math. Soc. 49 (1943), 923{931. [3] HOZO, I. and MILLER, H. I.: On Riemann's theorem about conditionally convergent series, Mat. Vesnik 38 (1986), 279{283. [4] KAHANE, J. P.: Some Random Series of Functions, Cambridge University Press, [5] Keagy, T. A.: Summability of certain category two classes, Houston Journ. of Math. 3 (1977), 61{65. [6] KEOGH, F. R. and PETERSEN, G. M.: A universal Tauberian theorem, Journ. London Math. Soc. 33 (1958), 121{123. [7] MILLER, H. I.: Generalization of a result of Borwein and Ditor, Proc. Amer. Math. Soc. 105 (1989), 889{893. [8] OXTOBY, J. C.: Measure and Category, Springer{Verlag, New York, [9] SZ USZ, P.: On a theorem of Buck and Pollard, Z. Wahrscheinlichkeitstheorie verw. Gebiete 11 (1968), 39{40.

The Degree of the Splitting Field of a Random Polynomial over a Finite Field

The Degree of the Splitting Field of a Random Polynomial over a Finite Field John D. Dixon and Daniel Panario School of Mathematics and Statistics Carleton University, Ottawa, Canada fjdixon,danielg@math.carleton.ca