In this paper we give incidence bounds for arrangements of curves in Fq . As an application, we prove a new result that if (x, f(x)) is a Sidon set then either A+A or f(A)+ f(A) should be large. The main goal of the paper is to illustrate the use of graph spectral techniques in additive combinatorics. This is an extended version of the talks I gave in the Additive Combinatorics DocCourse held a...

Many dynamic programming algorithms for discrete 0-1 optimization problems are just special (recursively constructed) tropical (min,+) or (max,+) circuits. A problem is homogeneous if all its feasible solutions have the same number of 1s. Jerrum and Snir [JACM 29 (1982), pp. 874– 897] proved that tropical circuit complexity of homogeneous problems coincides with the monotone arithmetic circuit ...

A symmetric subset of the reals is one that remains invariant under some reflection x 7→ c−x. Given 0 < ε ≤ 1, there exists a real number ∆(ε) with the following property: if 0 ≤ δ < ∆(ε), then every subset of [0, 1] with measure ε contains a symmetric subset with measure δ, while if δ > ∆(ε), then there exists a subset of [0, 1] with measure ε that does not contain a symmetric subset with meas...

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B2[g]-set is a set A of integers such that no integer has more than g essentially distinct representations as the sum of two elements of A. It is proved that almost all small sumsets of {1, 2, . . . , n} are B2[g]-s...

We construct integer error-correcting codes and covering for the limited-magnitude error channel with more than one error. The are lattices that pack or cover space appropriate ball. Some of constructions attain an asymptotic packing/covering density is constant. results obtained via various methods, including use in Hamming metric, modular B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" ...

We show that for any finite set A and an arbitrary $$\varepsilon >0$$ there exists $$k=k(\varepsilon )$$ such the higher energy $$\textsf{E} _k(A)$$ is at most $$|A|^{k+\varepsilon }$$ unless has a very specific structure. As application we obtain subset of real numbers or prime field either contains additive Sidon-type size $$|A|^{1/2+c}$$ multiplicative .