The fractional parts of the Bernoulli numbers are dense in the interval (0, 1). For every positiv... more The fractional parts of the Bernoulli numbers are dense in the interval (0, 1). For every positive integer k, the set of all m for which B2m has the same fractional part as B2R has positive asymptotic density.
Mathematical Statistics and Probability Theory, 1987
Consider a sequence ;h \-) of positive integers and investigate the properties of the maximal inc... more Consider a sequence ;h \-) of positive integers and investigate the properties of the maximal increments I(N .Iiv). This problem was studied by many authors in case of different { \}'s. In the present paper we intend to summarize the results and prove a few new theorems. We are especially interested in the case K\ = Iog A +->flo~-.~). In section 1 we introduce a few notations and concepts and recall the known results in the case Iív < olog N. In section 2 a key-inequality will be proved. The main results are presented in section 3. Section 4 gives a survey of the case log N « Ii v < .N. 1. NOTIONS AND A FEW KNOWN RESULTS In order to present the results of our paper in a pleasant form, it is uorthwile to recall some definitions, see, e .g. Révész (1980,1982). Let { _ ;7") be a sequence of r .v .'s. Then we formulate : DEFINITION 1. The sequence f&i) (n = 1 .2. .. .) belongs to the upper-upper class of S (f, c ü :,t'(t)) i# 7 .,, _< ,fl(n) a .s. for all but finitely many n.. DEFINITION 2. The sequence f2(n) (n = 1 .2. .. .) belongs to the upper-lower class of S (f 2 E ? C. r ,(S)) if Z" > ,f-,(n) a .s. i .o. DEFINITION 3. The sequence f3(") (ia = 1 .2. .. .) belongs to the lower-upper class of (U3 E:~ü :(C)) if Z" <,f3(n) a. s. i .o. DEFINITION 4. The sequence f,(n) (o = 1 .2. .. .) belongs to the lower-lower class of , (f, E _ (ti)) if Z" >_ f4(n) a .s. for all but finitely many n .
Let (;?l (n) = (;?(n) where (;? is Euler's function, let (;?2(n) = (;?((;?(n)) , etc. We prove se... more Let (;?l (n) = (;?(n) where (;? is Euler's function, let (;?2(n) = (;?((;?(n)) , etc. We prove several theorems about the normal order of (;?k(n) and state some open problems. In particular, we show that the normal order of (;?k(n)/(;?k+l(n) is ke"Ylogloglogn where I is Euler's constant. We also show that there is some positive constant c such that for all n, but for a set of asymptotic density 0 , there is some k with (;?k(n) divisible by every prime up to (logn)c. With k(n) the first subscript k with (;?k(n) = 1 , we show, conditional on a certain form of the Elliott-Halberstam conjecture, that there is some positive constant 0: such that k(n) has normal order 0: log n. Let s(n) = 0'(n)-n where 0' is the sum of the divisors function, let s2(n) = s(s(n)) ,etc. We prove that s2(n)/s(n) = s(n)/n+o(l) on a set of asymptotic density 1 and conjecture the same is true for sk+1(n)/sk(n) for any fixed k .
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1985
A lower limit of the length of the longest excursion of a symmetric random walk is given. Certain... more A lower limit of the length of the longest excursion of a symmetric random walk is given. Certain related problems are also discussed. It is shown e.g. that for any 5>0 and all sufficiently large n there are c(e) loglogn excursions in the interval (0, n) with total length greater than n(1-5), with probability 1.
Journal of the Australian Mathematical Society, 1979
We consider some unconventional partition problems in which the parts of the partition are restri... more We consider some unconventional partition problems in which the parts of the partition are restricted by divisibility conditions, for example, partitions n = a1+…+ak into positive integers a1, …, ak such that a1 ∣ a2 ∣ … ∣ ak. Some rather weak estimates for the various partition functions are obtained.
Let A be an infinite sequence of positive integers a, < a, < .I. and put fAxI = CnsR,oCx (l/a), D... more Let A be an infinite sequence of positive integers a, < a, < .I. and put fAxI = CnsR,oCx (l/a), D,(x) = maxtcnGx CoeA,o,n 1. In Part I, it was proved that limx++m supD,(x)/f,(x) = +co. In this paper, this theorem is sharpened by estimating DA(x) in terms of f,(x). It is shown that lim X+ +a3 sup DA(x) exp(-c,(logf,(x))') = +a~ and that this assertion is not true if c, is replaced by a large constant c2.
Let r(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investi... more Let r(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G : the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f(n, p), defined to be the minimum of t(G) over all connected graphs with n vertices and n-1 +p edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, p) is surprisingly small ; in particular ,fin, (n)=2loglogn+OfogloglogP) for any constant c>0, and fin, ti"T) = 2log(I + 1 ~),)± 4 for 0 <,,, < 1 and n sufficiently large. Bounds on t(G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t(G) from the independence number a(G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r-I vertices. 986 Academic Press, mG 1. INTRODUCTION, DEFINITIONS AND MAIN RESULTS Let t(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G : the number of vertices and edges, the radius, independence number, maximum clique size and connectivity .
Denote by G(n; m) a graph of n vertices and m edges. We prove that every G(n; [n2/4]-t I) contain... more Denote by G(n; m) a graph of n vertices and m edges. We prove that every G(n; [n2/4]-t I) contains a circuit of 2 edges for every 3 5 I < czn, also that every G(n; [ns/4] + 1) contams a k&, u.) with u,, = [cl log n] (for the definition of k&,,. II,) see the introduction). Finally for t > ro every G(n; [tn"z]) contains a circuit of 21 edges for 2 5 I < ~$2.
The solvability of the equation ala 2 • • • a k = x 2, al, a 2 ..... a k e ~ is studied for fixed... more The solvability of the equation ala 2 • • • a k = x 2, al, a 2 ..... a k e ~ is studied for fixed k and 'dense' sets M of positive integers. In particular, it is shown that if k is even and k I> 4, and M is of positive upper density, then this equation can be solved.
The cardinality of the finite set Y is denoted by ISI-.s& B8,. I s den&e finite or infinite sets ... more The cardinality of the finite set Y is denoted by ISI-.s& B8,. I s den&e finite or infinite sets of positive integers. If & is a finite or infinite set of positive integers, then S(d) denotes the set of the distinct positive integers n that can be represented in the form n = Coed &*a where E, = 0 or 1 for all a (and if JZZ is infinite, then all but finitely many of the E'S are equal to 0).
We consider a number of density problems for integer sequences with certain divisibility properti... more We consider a number of density problems for integer sequences with certain divisibility properties and sequences free of arithmetic progressions. Sequences of the latter type that are generated by a computer using modifications of the greedy algorithm are also provided. (~) 1999 Elsevier Science B.V. All rights reserved
It is known that for sufficiently large n and m and any r the binomial coefficient (~) which is c... more It is known that for sufficiently large n and m and any r the binomial coefficient (~) which is close to the middle coefficient is divisible by pr where p is a 'large' prime. We prove the exact divisibility of (,~) by p' for p>c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients. (~) 1999 Elsevier Science B.V. All rights reserved
Lower bounds on the Ramsey number r(G, H), as a function of the size of the graphs G and H, are d... more Lower bounds on the Ramsey number r(G, H), as a function of the size of the graphs G and H, are determined. In particular, if H is a graph with n lines, lower bounds for r(H) = r(H, H) and r(K,, H) are calculated in terms of n in the first case and m and n in the second case. For m = 3 an upper bound is also determined. These results partially answer a question raised by Harary about the relationship between Ramsey numbers and the size of graphs.
We study the distribution of lattice points a + ib on the fixed circle a 2 + b 2 = n. Our results... more We study the distribution of lattice points a + ib on the fixed circle a 2 + b 2 = n. Our results apply p.p. to the representable integers n, and we supply bounds for the discrepancy of the distribution, and for the maximum and minimum of the angles between consecutive points. As a corollary, we are able to show that when n is representable then it is almost surely representable with min(a,b) small, with an explicit bound. (~)1999 Elsevier Science B.V. All rights reserved
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