43 research outputs found
Problems and Results on Additive Properties of General Sequences IV
Let A={a1,a2,⋯}, a1<a2<⋯, be an infinite sequence of positive integers. One defines its counting function as A(n)=Card{a∈A;a≤n} (n=0,1,2,⋯), and the representation functions R1(n), R2(n), R3(n) (n=0,1,2,⋯), as being the number of representations of n in the form: (1) n=a+a′, a∈A, a′∈A, (2) n=a+a′, a<a′, a∈A, a′∈A, (3) n=a+a′, a≤a′, a∈A, a′∈A, respectively. Of course, for any n≥0, R1(n)=R2(n)+R3(n), and R3(n) is equal either to R2(n)+1, if n is even and n/2 belongs to A, or to R2(n), otherwise. In the first three parts of this series of papers [Erdős and Sárközy, Part I, Pacific J. Math. 118 (1985), no. 2, 347–357; MR0789175 (86j:11015); Part II, Acta Math. Hungar. 48 (1986), 201–211; MR0858398 (88c:11016); Part III, the authors, Studia Sci. Math. Hungar. 22 (1987), no. 1, 53–63], regularity properties of the asymptotic behavior of the function R1 were studied. In Parts IV and V the authors study monotonicity properties of the three functions R1,R2,R3.
In Part IV, they prove first that the function R1 is monotone increasing from a certain point on (i.e., there exists an n0 withR1(n+1)≥R1(n) for n≥n0) if and only if the sequence A contains all the integers from a certain point on. The proof uses elementary but complex considerations on counting functions. Secondly, they show that R2 has a different behavior, by exhibiting a class of sequences A satisfying A(n <n−cn1/3 for all large n and such that R2 is monotone increasing from some point onwards. The third result proved in Part IV is that if A(n)=o(n/logn) then the functions R2 and R3 cannot be monotone increasing from a certain point on. Here, the proof is based on analytic properties of the generating function f(z)=∑a∈Aza (|z|<1), corresponding to the sequence A. Part V treats the monotonicity of R3. The main result is as follows: If (4) limn→+∞(n−A(n))/logn=+∞, then lim supN→+∞∑k=1N(R3(2k)−R3(2k+1))=+∞. (Thus, roughly speaking, ai+aj assumes more even values than odd ones.) This theorem implies, firstly, that under hypothesis (4), which is weaker than A(n)=o(n/logn), R3 cannot be monotone increasing from a certain point on, and, secondly, that if A is an infinite "Sidon sequence'' (also called a "B2-sequence'', i.e., a sequence such that R3(n)≤1 for all n), then there are infinitely many integers k such that 2k can be represented in the form 2k=a+a′, a∈A, a′∈A, but 2k+1=a+a′, a∈A, a′∈A, is impossible. Part V finishes with the construction of a sequence showing that the main result is almost best possible. The proofs in Part V are of the same nature as those in Part IV
Enumerating Abelian Returns to Prefixes of Sturmian Words
We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian
returns in Sturmian words. We determine the cardinality of the set
of abelian returns of all prefixes of a Sturmian word in
terms of the coefficients of the continued fraction of the slope, dependingly
on the intercept. We provide a simple algorithm for finding the set
and we determine it for the characteristic Sturmian words.Comment: 19page
The critical window for the classical Ramsey-Tur\'an problem
The first application of Szemer\'edi's powerful regularity method was the
following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any
K_4-free graph on N vertices with independence number o(N) has at most (1/8 +
o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising
geometric construction, utilizing the isoperimetric inequality for the high
dimensional sphere, of a K_4-free graph on N vertices with independence number
o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in
1976, several problems have been asked on estimating the minimum possible
independence number in the critical window, when the number of edges is about
N^2 / 8. These problems have received considerable attention and remained one
of the main open problems in this area. In this paper, we give nearly
best-possible bounds, solving the various open problems concerning this
critical window.Comment: 34 page