7 research outputs found
Explicit constructions of infinite families of MSTD sets
We explicitly construct infinite families of MSTD (more sums than
differences) sets. There are enough of these sets to prove that there exists a
constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD
sets; thus our family is significantly denser than previous constructions
(whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We
conclude by generalizing our method to compare linear forms epsilon_1 A + ... +
epsilon_n A with epsilon_i in {-1,1}.Comment: Version 2: 14 pages, 1 figure. Includes extensions to ternary forms
and a conjecture for general combinations of the form Sum_i epsilon_i A with
epsilon_i in {-1,1} (would be a theorem if we could find a set to start the
induction in general
Binary linear forms over finite sets of integers
Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with
integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this
paper it is proved that for every pair of normalized binary linear forms
f(x,y)=u_1x+v_1y and g(x,y)=u_2x+v_2y with integral coefficients, there exist
arbitrarily large finite sets of integers A and B such that |f(A)| > |g(A)| and
|f(B)| < |g(B)|.Comment: 20 page
In Vitro Cell Culture Infectivity Assay for Human Noroviruses
A 3-dimensional organoid human small intestinal epithelium model was used