A System of Two Diophantine Inequalities with Primes

Let 1 < d < c < 128 / 119 , 1 < α < β < 6 1 − d / c . In this paper, we prove that there exist positive real numbers N 1 0 and N 2 0 depending on c , d , α , β such that for all real numbers N 1 > N 1 0 , N 2 > N 2 0 and α ≤ N 2 / N 1 d / c ≤ β , the system of two Diophantine inequalities p 1 c + ⋯ + p 6 c − N 1 < N 1 − 1 / c 128 / 119 − c log 109 N 1 , p 1 d + ⋯ + p 6 d − N 2 < N 2 − 1 / d 128 / 119 − d log 109 N 2 is solvable in prime variables p 1 , … , p 6 .

Diophantine Inequalities for Generic Ternary Diagonal Forms

Let $k\geq 2$ and consider the Diophantine inequality $$ \left|x_1^k-{\alpha}_2 x_2^k-{\alpha}_3 x_3^k\right| &lt;{\theta}.$$Our goal is to find non-trivial solutions in the variables $x_i$, $1\leq i\leq 3$, all of size about $P$, assuming that ${\theta }$ is sufficiently large. We study this problem on average over ${\alpha }_3$ and generalize previous work by Bourgain on quadratic ternary diagonal forms to general degree $k$.

Proportionally modular affine semigroups

This work introduces a new kind of semigroup of [Formula: see text] called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets, and we specialize when [Formula: see text]. For this case, we also provide a faster algorithm to compute their minimal system of generators, prove they are Cohen–Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.