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Distribution of lattice points on hyperbolic surfaces

Vsevolod F. Lev — 1996

Acta Arithmetica

Let two lattices $Λ^{'}, Λ^{''} \subset ℝ^{s}$ have the same number of points on each hyperbolic surface $| x ₁ . . . x_{s} | = C$ . We investigate the case when Λ’, Λ” are sublattices of $ℤ^{s}$ of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.

Restricted set addition in Abelian groups: results and conjectures

Vsevolod F. Lev — 2005

Journal de Théorie des Nombres de Bordeaux

We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdős-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed. The paper assumes no expertise from the reader and can serve as an introduction to the subject.

Powers of 2 with five distinct summands

Vsevolod F. Lev — 2008

Acta Arithmetica

Blocks and progressions in subset sum sets

Vsevolod F. Lev — 2003

Acta Arithmetica

On addition of two distinct sets of integers

Vsevolod F. Lev; Pavel Y. Smeliansky — 1995

Acta Arithmetica

What is the structure of a pair of finite integers sets A,B ⊂ ℤ with the small value of |A+B|? We answer this question for addition coefficient 3. The obtained theorem sharpens the corresponding results of G. Freiman.

Solving a ± b = 2c in elements of finite sets

Vsevolod F. Lev; Rom Pinchasi — 2014

Acta Arithmetica

We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates...

Character sums in complex half-planes

Sergei V. Konyagin; Vsevolod F. Lev — 2004

Journal de Théorie des Nombres de Bordeaux

Let $A$ be a finite subset of an abelian group $G$ and let $P$ be a closed half-plane of the complex plane, containing zero. We show that (unless $A$ possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of $A$ which belongs to $P$ . In other words, there exists a non-trivial character $χ \in \hat{G}$ such that $\sum_{a \in A} χ (a) \in P$ .

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Currently displaying 1 – 12 of 12

Distribution of lattice points on hyperbolic surfaces

Restricted set addition in Abelian groups: results and conjectures

Powers of 2 with five distinct summands

Blocks and progressions in subset sum sets

On addition of two distinct sets of integers

Solving a ± b = 2c in elements of finite sets

Character sums in complex half-planes

Distribution of points on arcs.

Restricted set addition in groups. II: A generalization of the Erdős-Heilbronn conjecture.

Reconstructing integer sets from their representation functions.

On the distribution of exponential sums.

On the size of dissociated bases.