2017/03/02 by David Covert, Covert, David, Doowon Koh +3 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #math.CA #math.CO
paper · pdf · doi:10.48550/arxiv.1703.00609
arxiv created 2017/07/17 · arxiv updated 2017/07/18
Let \mathbb Fqd be the d-dimensional vector space over the finite field \mathbb Fq with q elements. Given k sets Ej⊂ \mathbb Fqd for j=1,2,…, k, the generalized k-resultant modulus set, denoted by Δk(E1,E2, …, Ek), is defined by Δk(E1,E2, …, Ek)=\‖\bf x1+\bf x2+⋯+\bf xk‖∈ \mathbb Fq:\bf xj∈ Ej, j=1,2,…, k\, where ‖\bf y‖=\bf y12+ ⋯ + \bf yd2 for \bf y=(\bf y1, …, \bf yd)∈ \mathbb Fqd. We prove that if ∏j=13 |Ej| ≥ C q3((d+1)/(2) -(1)/(6d+2)) for d=4,6 with a sufficiently large constant C>0, then |Δ3(E1,E2,E3)|≥ cq for some constant 0<c≤ 1, and if ∏j=14 |Ej| ≥ C q4((d+1)/(2) -(1)/(6d+2)) for even d≥ 8, then |Δ4(E1,E2,E3, E4)|≥ cq. This generalizes the previous result in \citeCKP16. We also show that if ∏j=13 |Ej| ≥ C q3((d+1)/(2) -(1)/(9d-18)) for even d≥ 8, then |Δ3(E1,E2,E3)|≥ cq. This result improves the previous work in \citeCKP16 by removing ε>0 from the exponent.