On distinct perpendicular bisectors and pinned distances in finite fields

On distinct sums and distinct distances

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all ð 2Þn sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least ns ; where ds 1⁄4 1=cJs=2n is defined in...

Pinned synthetic ferrimagnets with perpendicular anisotropy and tuneable exchange bias

On Pinned Fields, Interlacements, and Random Walk on (z/nz)

We define two families of Poissonian soups of bidirectional trajectories on Z2, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus (Z/NZ)2, started from the uniform distribution, run up to a time of order (N logN)2 and forced to avoid a fixed point. The local limit of the latter was recently established in [6]. Our construc...

Distinct Distances on a Sphere

We prove that a set of N points on a two dimensional sphere satisfying a discrete energy condition determines at least a constant times N distinct distances. Homogeneous sets in the sense of Solymosi and Vu easily satisfy this condition, as do other sets that in the sense that will be made precise below respect the curvature properties of the sphere. The classical Erdös distance conjecture (EDC...

A note on distinct distances

We show that, for a constant-degree algebraic curve γ in R, every set of n points on γ spans at least Ω(n) distinct distances, unless γ is an algebraic helix (see Definition 1.1). This improves the earlier bound Ω(n) of Charalambides [2]. We also show that, for every set P of n points that lie on a d-dimensional constantdegree algebraic variety V in R, there exists a subset S ⊂ P of size at lea...