A Structure result for bricks in Heisenberg groups

We show that for a sufficiently big \textit{brick} $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field $\mathbb{F}_p$, the product set $B\cdot B$ contains at least $|B|/p$ many cosets of some non trivial subgroup of $H_n$

On arithmetic sums of Cantor-type sequences of integers

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate $FS(B)$ the subset sum of sequence type $B=\{\lfloor p^n\alpha\rfloor\}^\infty_{n=0}$. When $p=2$, then we prove $FS(B)+FS(B)=\N$ by analogy with the Cantor set, and some structure theorem for $p>2$Comment: comments are welcom

Estimation of function's supports under arithmetic constraints

The well-known $|supp(f)||supp(\widehat{f}|\geq |G|$ inequality gives lower estimation of each supports. In the present paper we give upper estimation under arithmetic constrains. The main notion will be the additive energy which plays a central role in additive combinatorics. We prove an uncertainty inequality that shows a trade-off between the total changes of the indicator function of a subset $A\subseteq \mathbb F^n_2$ and the additive energy of $A$ and the Fourier spectrum.Comment: 8 pages, Submitted to the journal "ADAM" in December 202

A note on Freiman models in Heisenberg groups

Green and Ruzsa recently proved that for any $s\ge2$, any small squaring set $A$ in a (multiplicative) abelian group, i.e. $|A\cdot A|<K|A|$, has a Freiman $s$-model: it means that there exists a group $G$ and a Freiman $s$-isomorphism from $A$ into $G$ such that $|G|<f(s,K)|A|$. In an unpublished note, Green proved that such a result does not necessarily hold in non abelian groups if $s\ge64$. The aim of this paper is improve Green's result by showing that it remains true under the weaker assumption $s\ge6$

Iterated compositions of linear operations on sets of positive upper density

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density

On subset sums of pseudo–recursive sequences

Let a0 = a 2 N, fMig1i =1 be an infinite set of integers and fb1; b2; : : : ; bkg be a finite set of integers. We say that faig1 i=0 is a pseudo-recursive sequence if an+1 = Mn+1an + bjn+1 (bjn+1 2 fb1; b2; : : : bkg) holds. In the first part of the paper, we investigate the subset sum of a generalized version of A := fan = b2nc : n = 0; 1; 2; : : : g, which is a special pseudorecursive sequence. In the second part, we use A for an encryption algorith

Additív és multiplikatív számelmélet = Additive and multiplicative number theory

1. A Goldbach-sejtésről: Ismert a páros Golbach-sejtéssel kapcsolatban, hogy majdnem minden páros szám esetén igaz a páros Goldbach-sejtés. A pályázat keretei között a korábbi eredményeket Pintz János lényegesen megjavította, megmutatva, hogy a kivételek száma X-ig legfeljebb O(X^{2/3}). A páros Goldbach-sejtés irányában Linnik bizonyította, hogy van olyan K korlát, hogy bármely elég nagy páros szám előáll két prímszám és legfeljebb K darab 2-hatvány összegeként. Pintz János és Ruzsa Imre kutatásai eredményeként sikerült csökkenteni a K-ra a felső korlátot (K=8). 2. Konzekutív prímszámok hézagairól Jelölje d_k a (k+1)-edik és a k-ik prímszám differenciáját. Pintz János, Goldstonnal és Yildirimmel a szitamódszereket továbbfejlesztve, bebizonyította, hogy d_k végtelen sok k esetén legfeljebb (log k)^{1/2}(log log k)^2 nagyságrendű. 3. Néhány diofantikus egyenletről Balog és Ono elliptikus görbékhez hozzárendelt algebrai objektumok, nevezetesen az ideálosztályok csoportja és a Safarevics-Tate csoport szerkezetével foglalkoztak. Ez a 2m^k=p_1+p_2 egyenlet megoldhatóságára vezet, ahol p_1 és p_2 prímek. 4. Hilbert-kockákkal kapcsolatos kérdések. Hegyvári N. és Sándor Cs. Hilbert kockák dimenziójára és különböző halmazokban való előfordulását vizsgálja. 5. Freiman-típusú kérdések Elekes György és Ruzsa Imre vizsgálta kicsi összeghalmazok Freiman-féle elméletét arra az esetre, amikor nem az összes összeget képezzük, hanem egy adott gráfban összekötött szám-párokat adjuk össze. | 1. On Goldbach-conjecture The exceptional Goldbach-set is defined as a set of all even numbers which cannot be written as a sum of two prime numbers. The best known result is O(X^{0.92}) up to X. During the period of our project Pintz improved it to O(X^{2/3}). As an approximation to the Goldbach conjecture Linnik examined the problem that for which K will be true that every large integer is a sum of two primes and at most K powers of two. Using the GRH Pintz and Ruzsa proved that K is at most 7, and K is at most 8 without any hypothesis. The best known results were 2250 and 200 respectively. 2. On consecutive gaps of the sequence of prime numbers Let d_k be the gap between the k^th and the (k+1)^th primes. Pintz (with Goldston és Yildirim) proved that d_k<c(log k)^{1/2}(log log k)^2 for infinitely many k. 3. On a diofantine equation The paper of A. Balog and K. Ono studies the structure of class group of imaginary quadratic fields. The study is closely connected to the solvability of 2m^k=p_1+p_2, where k is fixed, m is an integer and p_1, p_2 are primes. 4. On Hilbert cubes Hegyvári and Sándor investigated the dimension and the structure of Hilbert cubes in certain sets. 5. Freiman-type questions Let A be a finite set and let G be a finite graph on the vertices {1,2,?card(A)}. A restricted addition A+_G A is the set of all elements which can be written as a_i +a_j if and only if (i,j) in E(G). Elekes and Ruzsa proved that if G is strongly connected then the sumset is 'small'

Additive structure of difference sets and a theorem of Følner

A theorem of Folner asserts that for any set A subset of Z of positive upper density there is a Bohr neigbourhood B of 0 such that B \ (Lambda - Lambda) has zero density. We use this result to deduce some consequences about the structure of difference sets of sets of integers having a positive upper density