Roth's Theorem Research Articles

Quantitative bounds in the nonlinear Roth theorem

Quantitative bounds in the nonlinear Roth theorem

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

Two-Point Polynomial Patterns in Subsets of Positive Density in ℝn

Abstract Let $\gamma (t)=(P_{1}(t),\ldots ,P_{n}(t))$ where $P_{i}$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+\gamma (t)\}$ in sets of positive density $\epsilon $ in $[0,1]^{n}$ with a gap estimate $t\geq \delta (\epsilon )$ when $P_{i}$’s are arbitrary, and in $[0,N]^{n}$ with a gap estimate $t\geq \delta (\epsilon )N^{n}$ when $P_{i}$’s are of distinct degrees where $\delta (\epsilon )=\exp \left (-\exp \left (c\epsilon ^{-4}\right )\right )$ and $c$ only depends on $\gamma $. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain’s reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.

Roth's theorem and the Hardy–Littlewood majorant problem for thin subsets of primes

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy–Littlewood majorant property holds for these subsets of primes. Notably, our considerations recover the results for the Piatetski–Shapiro primes for exponents close to 1, which are primes of the form ⌊nc⌋ for a fixed c>1.

A polynomial Roth theorem for corners in $${\mathbb {R}}^2$$ and a related bilinear singular integral operator

A polynomial Roth theorem for corners in $${\mathbb {R}}^2$$ and a related bilinear singular integral operator

Tower-type bounds for Roth’s theorem with popular differences

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every \epsilon > 0 there is some N_0(\epsilon) such that for every N \ge N_0(\epsilon) and A \subset [N] with |A| = \alpha N , there is some nonzero d such that A contains at least (\alpha^3 - \epsilon) N three-term arithmetic progressions with common difference d . We prove that the minimum N_0(\epsilon) in Green's theorem is an exponential tower of twos of height on the order of \log(1/\epsilon) . Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.

Irregularities of distribution and geometry of planar convex sets

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When ∂C is C2 regardless of curvature, we prove that for every set PN of N points in T2 we have the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN1/2. When ∂C is only piecewise C2 and is not a polygon we prove the sharp bound∫01∫T2|card(PN∩(τC+t))−τ2N|C||2dtdτ⩾cN2/5. We also give a whole range of intermediate sharp results between N2/5 and N1/2. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.

A simple and constructive proof to a generalization of Lüroth's theorem

A generalization of L{\"u}roth's theorem expresses that every transcendence degree 1 subfield of the rational function field is a simple extension. In this note we show that a classical proof of this theorem also holds to prove this generalization.

An uncountable ergodic Roth theorem and applications

<p style='text-indent:20px;'>We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-systems for uniformly amenable groups <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. Our uncountable Roth theorem is crucial in the proof of both of these results.</p>

Roth’s Theorem over arithmetic function fields

Roth's theorem is extended to finitely generated field extensions of $\Bbb Q$, using Moriwaki's framework for heights.

Improved bound in Roth's theorem on arithmetic progressions

We prove that if A⊆{1,…,N} does not contain any non-trivial three-term arithmetic progression, then|A|≪(log⁡log⁡N)3+o(1)log⁡NN.

Two bipolynomial Roth theorems in [formula omitted

We give two Roth theorems, related to the nonlinear configuration x, x+P1(t), x+P2(t) involving two polynomials, for sets in R of positive density and of fractional dimensions. The proof uses Fourier analysis.

Polynomial Roth Theorems on Sets of Fractional Dimensions

Abstract Let $E\subset{\mathbb{R}}$ be a closed set of Hausdorff dimension $\alpha \in (0, 1)$. Let $P: {\mathbb{R}}\to{\mathbb{R}}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Łaba and Pramanik [ 11] to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot et al. [ 7].

A general nonlinear version of Roth's theorem on the real line

A general nonlinear version of Roth's theorem on the real line

Szemerédi’s proof of Szemerédi’s theorem

In 1975, Szemeredi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemeredi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting densities of sets along multidimensional arithmetic progressions. In this note we present an arrangement of this proof that incorporates a number of notational and technical simplifications. Firstly, we replace the use of the regularity lemma by that of the simpler ``weak regularity lemma'' of Frieze and Kannan. Secondly, we extract the key inductive steps at the core of Szemeredi's proof (referred to as ``Lemma 5'', ``Lemma 6'', and ``Fact 12'' in that paper) as stand-alone theorems that can be stated with less notational setup than in the original proof, in particular involving only (families of) one-dimensional arithmetic progressions, as opposed to multidimensional arithmetic progressions. Thirdly, we abstract the analysis of limiting densities along the (now one-dimensional) arithmetic progressions by introducing the notion of a family of arithmetic progressions with the ``double counting property''. We also present a simplified version of the argument that is capable of establishing Roth's theorem on arithmetic progressions of length three.

Lacunary Power Series and 𝑼𝒎-Numbers

AbstractIn this work, the values of certain lacunar power series with rational coefficients for 𝑈𝒎-number arguments were determined to be either in a particular algebraic number field or in the set of transcendental numbers under specific circumstances in the complex numbers field. The result was also applied on some of the lacunary power series with coefficients in an algebraic number field. Roth's theorem which is the essential result in Diophantine approximation to algebraic numbers was used to reach the present results.

A polynomial Roth theorem on the real line

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

An analytic approach to sparse hypergraphs: hypergraph removal

An analytic approach to sparse hypergraphs: hypergraph removal, Discrete Analysis 2018:3, 47 pp. The famous triangle removal lemma of Ruzsa and Szemerédi states that for every $\epsilon>0$ there exists $\delta>0$ such that every graph $G$ on $n$ vertices with at most $\delta n^3$ triangles has a triangle-free subgraph $G'$ such $|E(G)\setminus E(G')|\leq\epsilon n^2$. To put it less formally, a graph that is almost triangle free can be approximated by a graph that is exactly triangle free. A surprising application of this innocent looking result is a proof of Roth's theorem, that for every $\delta>0$ there exists $n$ such that every subset $A\subset\{1,2,\dots,n\}$ of size at least $\delta n$ contains an arithmetic progression of length 3. To prove this deduction, one defines a tripartite graph with appropriate vertex sets $X,Y,Z$ (one can, for instance, take each set to be $\{1,2,\dots,3n\}$), with $xy$ an edge if $y-x\in A$, $yz$ an edge if $z-y\in A$, and $xz$ an edge if $(z-x)/2\in A$. Then a triangle $xyz$ in the graph corresponds to an arithmetic progression $y-x,z-x,z-y$ in $A$ (which may be degenerate, but there cannot be too many degenerate ones). This lemma has been generalized in several different directions. One is to a "simplex removal lemma" for hypergraphs. For 3-uniform hypergraphs, for example, it states that for every $\epsilon>0$ there exists $\delta>0$ such that if a 3-uniform hypergraph $H$ on $n$ vertices contains at most $\delta n^4$ simplices (that is, configurations of the form $xyz,xyw,xzw,yzw$), then one can throw away at most $\epsilon n^3$ triples from $H$ in order to obtain a subhypergraph $H'$ that contains no simplices at all. Another direction of generalization is to a "sparse random version". Here the idea is to start with a random graph $U$ with $n$ vertices and edge probability $n^{-\alpha}$ for some suitable $\alpha>0$, and to prove a result about subgraphs $G$ of $U$, where the various densities are measured relative to $U$. Thus, $G$ is deemed to have few triangles if the number of triangles in $G$ is at most $\delta$ times the number of triangles in $U$, and $G'$ is deemed to be a good approximation of $G$ if it differs from $G$ by at most $\epsilon$ times the number of edges in $U$. Of course, one can try to generalize the result in both directions simultaneously in order to obtain a sparse random version of the simplex removal lemma. The purpose of this paper is to give a new way of proving such a generalization. The methods used are infinitary, continuing a line of research initiated by Lovász and various coauthors with their introduction of graph and hypergraph limits. The novelty of the paper is that it manages to apply an infinitary approach of this kind in the sparse random context, which was not obviously possible and which required the author to work out how to deal with some serious-looking obstacles.

Long-time existence for the semilinear Klein–Gordon equation on a compact boundary-less Riemannian manifold

ABSTRACTWe investigate the long-time existence of small and smooth solutions for the semilinear Klein–Gordon equation on a compact boundary-less Riemannian manifold. Without any spectral or geometric assumption, our first result improves the lifespan obtained by the local theory. The previous result is proved under a generic condition of the mass. As a by-product of the method, we examine the particular case, where the manifold is a multidimensional torus, and we give explicit examples of algebraic masses for which we can improve the local existence time. The analytic part of the proof relies on multilinear estimates of eigenfunctions and estimates of small divisors proved by Delort–Szeftel. The algebraic part of the proof relies on a multilinear version of the Roth theorem proved by Schmidt.