Exceptional sets in Waring's problem: two squares and s biquadrates

Bounds on squares of two-sets

Exceptional Sets and Fiber Products

Exceptional sets where fibers have dimensions higher than the generic fiber dimension are of interest in mathematics and in application areas, such as the theory of overconstrained mechanisms. We show that fiber products promote such sets to become irreducible components, whereupon they can be found using techniques from numerical algebraic geometry for computing the irreducible decomposition. ...

Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n log n) time, where n denotes the total number of points. We also show that the analogous...

Squares and Difference Sets in Finite Fields

For infinitely many primes p = 4k+1 we give a slightly improved upper bound for the maximal cardinality of a set B ⊂ Zp such that the difference set B−B contains only quadratic residues. Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B| ≤ √ p − 1, under suitable conditions on p. The new bound is valid for approximately three quarters of the primes p = 4k + 1.

On the Waring–goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers

(1.1) p1 + · · ·+ ps = n, where p1, . . . , ps are prime unknowns. It is conjectured that for any pair of integers k, s ∈ N with s ≥ k + 1 there exist a fixed modulus qk,s and a collection Nk,s of congruence classes mod qk,s such that (1.1) is solvable for all sufficiently large n ∈ Nk,s. While a proof of this conjecture appears to be beyond the reach of present methods, some significant progre...