Related papers: Solution-free sets for sums of binary forms

We prove that for any fixed integer \( n \geq 3 \) and nonzero integer \( m \), the proportion of integral binary forms of degree \( n \) that represent \( m \) tends to zero as the height tends to infinity. In fact, almost all such forms…

We derive asymptotic bounds for the ordinary generating functions of several classical arithmetic functions, including the Moebius, Liouville, and von Mangoldt functions. The estimates result from the Korobov-Vinogradov zero-free region for…

In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and…

We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace…

The density of a subgroupoid with respect to a free groupoid is defined as the asymptotic ratio of their growths. This notion can be interpreted as a generalisation of the index's inverse for groups or as the probability of an element…

In this paper, we find an expression for the density of the sum of two independent $d-$dimensional Student $t-$random vectors $\mathbf{X}$ and $\mathbf{Y}$ with arbitrary degrees of freedom. As a byproduct we also obtain an expression for…

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the…

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2x2x...x2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary…

We derive, via the Hardy-Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of non-singular local solubility. Our polynomials $F({\mathbf…

We determine the density of the largest sum-free subset of the lattice cube $\{1, 2, \dots, n\}^d$ for $d = 3$ and $d = 4$. This solves a conjecture of Cameron and Aydinian in dimensions $3$ and $4$.

We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in…

In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general…

For a singularly perturbed system of reaction--diffusion equations, assuming that the 0th order solutions in regular and singular regions are all stable, we construct matched asymptotic expansions for formal solutions to any desired order…

In this paper we study the asymptotic behaviour via Gamma-convergence of some integral functionals which model some multi-dimensional structures and depend explicitly on the linearized strain tensor. The functionals are defined in…

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size…

We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size using a…

We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we…

We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

The main objective of this article is to study the asymptotic behavior of Salie sums over arithmetic progressions. We deduce from our asymptotic formula that Salie sums possess a bias of being positive. The method we use is based on…