52 citations · 127 across the 37 of their papers we have counts for
9 papers · 1 filter
On distinct perpendicular bisectors and pinned distances in finite fields
Brandon Hanson, Ben Lund, Oliver Roche-Newton
Given a set of points such that it is established that determines distinct perpendicular bisectors. It is also proven tha…
Sets with few distinct distances do not have heavy lines
Orit E. Raz, Oliver Roche-Newton, Micha Sharir
Let be a set of points in the plane that determines at most distinct distances. We show that no line can contain more than points of $P…
A bound on the multiplicative energy of a sum set and extremal sum-product problems
Oliver Roche-Newton, Dmitry Zhelezov
In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information abo…
Elementary methods for incidence problems in finite fields
Javier Cilleruelo, Alex Iosevich, Ben Lund +2
We use elementary methods to prove an incidence theorem for points and spheres in . As an application, we show that any point set of with…
New sum-product type estimates over finite fields
Oliver Roche-Newton, Misha Rudnev, Ilya D. Shkredov
Let be a field with positive odd characteristic . We prove a variety of new sum-product type estimates over . They are derived from the theorem that the number of inciden…
Polynomial Values in Subfields and Affine Subspaces of Finite Fields
Oliver Roche-Newton, Igor Shparlinski
For an integer , a prime power , and a polynomial over a finite field of elements, we obtain an upper bound on the frequency of elements in an o…
Sum-ratio estimates over arbitrary finite fields
Oliver Roche-Newton
The aim of this note is to record a proof that the estimate holds for any set , provided that satisfies certai…
New sum-product estimates for real and complex numbers
Antal Balog, Oliver Roche-Newton
A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results…
Variations on the Sum-Product Problem
Brendan Murphy, Oliver Roche-Newton, Ilya D. Shkredov
This paper considers various formulations of the sum-product problem. It is shown that, for a finite set , g…