Geometric Structure of Sumsets
نویسنده
چکیده
In particular, the distance from a point x ∈ R to a hyperplane H where x / ∈ H , is given by the length of the perpendicular line segment from x to H . If two hyperplanes H1, H2 are parallel, their normal vectors are multiples of each other, so we can take a single normal vector u and write H1 = {x : (x, u) = α1} and H2 = {x : (x, u) = α2}. Take any x ∈ H1 . Then d(x,H2) is given by the perpendicular line segment. To calculate
منابع مشابه
Algebraic Proof for the Geometric Structure of Sumsets
We consider a finite set of lattice points and their convex hull. The author previously gave a geometric proof that the sumsets of these lattice points take over the central regions of dilated convex hulls, thus revealing an interesting connection between additive number theory and geometry. In this paper, we will see an algebraic proof of this fact when the convex hull of points is a simplex, ...
متن کاملThe additive structure of the squares inside rings
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set’s underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the min...
متن کاملOn Sets Free of Sumsets with Summands of Prescribed Size
We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.
متن کاملThe number of sumsets in a finite field
We prove that there are 2p/2+o(p) distinct sumsets A + B in Fp where |A|, |B| → ∞ as p →∞.
متن کاملSome Additive Combinatorics Problems in Matrix Rings
We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers. 2000 Mathematics Subject Classification. 11C20, 11D79, 11T23
متن کامل