Related papers: Roth-type theorems in $K_{s,t}$-free sets

The existence of higher derivative discontinuous solutions to a first order ordinary differential equation is shown to reveal a nonlinear SL(2,R) structure of analysis in the sense that a real variable $t$ can now accomplish changes not…

For smooth finite fields $F_q$ (i.e., when $q-1$ factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division,…

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal…

S\'ark\"ozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of S\'ark\"ozy's theorem…

De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a generator, and proved upper and lower bounds on the expected…

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…

There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical…

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of…

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$…

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free…

Let $R$ be a real smooth affine domain of dimension $3$ such that $R$ has either no real maximal ideals or the intersection of all real maximal ideals in $R$ has height at least $1$. Then we prove that all stably free $R$-modules of rank…

Let $n>2r>0$ be integers. We consider families $\mathcal{F}$ of subsets of an $n$-element set, in which the union of any two members has size at most $2r$. One of our results states that for $n\geq 6r$ the number of members of size…

Let $K=\mathbb{Q}(\omega)$ with $\omega$ the root of a degree $n$ monic irreducible polynomial $f\in\mathbb{Z}[X]$. We show the degree $n$ polynomial $N(\sum_{i=1}^{n-k}x_i\omega^{i-1})$ in $n-k$ variables formed by setting the final $k$…

A challenging problem is to find an algorithm to decide whether a morphism is k-power-free. We provide such an algorithm when k >= 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite…

Let \sigma be an automorphism of a commutative k-algebra R. The skew polynomial ring R[t;\sigma] is generated by R and an indeterminate t subject to the relations ta=\sigma(a)t for all a in R. For certain R and appropriate \sigma there are…

We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension…

Let $f,g \in k[x]$ be nonconstant polynomials over a number field $k$. We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\circ h$. The…

An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J.…

Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any…

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}=\{F_1,\cdots,F_m\} \subset…