Almost arithmetic progressions in the primes and other large sets

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.

The Riemann-zeta function on vertical arithmetic progressions

We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real, exhibits a remarkable correspondance with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression $a n + b$ are not the ordinates of some zero of $ζ(s)$ lying on the critical line. This improves on earlier work of Martin and Ng. We then complement this result by producing large and small values of $ζ(s…

Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions

In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $Δ>0$ with direction $e\in S^{d-1}$ inside the $εΔ$ neighbourhood of $F$ for all $ε>0$, $k\geq 3$ and $e\in S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $\mathbb{R}^d$ which asymptotically and om…

On Assouad dimension and arithmetic progressions in sets defined by digit restrictions

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any $0\le s\le 1$, there exists some set on $\mathbb{R}$ with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.

Sets with distinct sums of pairs, long arithmetic progressions, and continuous mappings

We show that if $φ\colon \mathbb R\rightarrow\mathbb R$ is a continuous mapping and the set of nonlinearity of $φ$ has nonzero Lebesgue measure, then $φ$ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.

Riesz sequences and generalized arithmetic progressions

The purpose of this note is to verify that the results attained in [6] admit an extension to the multidimensional setting. Namely, for subsets of the two dimensional torus we find the sharp growth rate of the step(s) of a generalized arithmetic progression in terms of its size which may be found in an exponential systems satisfying the Riesz sequence property.

欧几里得空间中避开无限等差数列的大子集

Large subsets of Euclidean space avoiding infinite arithmetic progressions

针对正测度集合必含任意长有限等差数列的结论，本文构造了在每个单位区间内测度至少为任意给定 $λ<1$、却不含任何无限等差数列的实数子集。

Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions

By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.