Distinct sums over subsets

Exponential Sums over Definable Subsets of Finite Fields

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application,...

Sums of distinct squares

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...

Distinct volume subsets

Suppose that a and d are positive integers with a ≥ 2. Let ha,d(n) be the largest integer t such that any set of n points in Rd contains a subset of t points for which all the nonzero volumes of the (t a ) subsets of order a are distinct. Beginning with Erdős in 1957, the function h2,d(n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h2,d...

Large Convexly Independent Subsets of Minkowski Sums

Let Ed(n) be the maximum number of pairs that can be selected from a set of n points in Rd such that the midpoints of these pairs are convexly independent. We show that E2(n) > Ω(n √ log n), which answers a question of Eisenbrand, Pach, Rothvoß, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (20...