24 problems

Let $\operatorname{E}$ be the intersection of all maximal Hardy fields, let $\operatorname{E}^\infty$ be the intersection of all maximal smooth Hardy fields, and let…

Maximal Hardy-field conjecture. If $H$ is maximal, then there is no $y\in\mathcal{C}^1$ such that

Let $k\in\mathbb{N}$ and let $a_1,\ldots,a_k\in\mathcal{H}$ be functions of polynomial growth. Assume that every non-trivial linear combination $a$ of $a_1,\ldots,a_k$ satisfies ……

Let $\mathcal{H}$ be the Hardy field under consideration, let $a\in\mathcal{H}$ have polynomial growth, and assume … for every $c\in\mathbb{R}$ and $p(t)\in\mathbb{Z}[t]$. Let…

Let $a_1,\ldots,a_k$ be functions from a Hardy field with polynomial growth, and consider averages along the primes whose iterates are given by $\lfloor a_i(p)\rfloor$. Frantzikina…

Let $(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system, and let … where $(c_i^1,c_i^2)\in\mathbb{R}^2\setminus\{(0,0)\}$, $h_i\in\mathcal{H}$, and $t_i\in\mathcal{T}$. Assume…

Conjectured optimal convergence criterion. Under this assumption, the averages

Hardy-field extension conjecture. Any Hardy field containing $mathbb R$ extends to a newtonian $omega$-free Hardy field.

Let $\mathcal{H}$ be a Hardy field, let $f_1,\ldots,f_k,W\in\mathcal{H}$, and define … Property (WP) is that, for every $f\in\operatorname{span}^*(f_1,\ldots,f_k)$ and…

Nilsequence structure conjecture. There exist a nilmanifold $Y=G/\Gamma$, a continuous function $F\in C(Y)$, a point $y\in Y$, elements $a_1,\ldots,a_k\in G$, and a sequence…

Let $f_1,\ldots,f_k$ be linearly independent functions of the form … Let $(X,\mathcal{B},\mu,T)$ be a totally ergodic measure-preserving system and let…

Let $f_1,\ldots,f_k$ be functions from a Hardy field, and define … Let $(X,\mathcal{B},\mu,T)$ be an ergodic measure-preserving system and let $h_1,\ldots,h_k\in L^\infty(X)$. Fran…

Frantzikinakis's conjecture. If $|f(t)-q(t)|\to\infty$ for every $f\in\operatorname{span}^*(f_1,\ldots,f_k)$ and $q\in\mathbb{Z}[t]$, then every set $E\subset\mathbb{N}$ of positiv…

Let $f_1,\ldots,f_k$ be functions of polynomial growth from a Hardy field. For every nonzero real linear combination … assume that … for every $p\in\mathbb{Z}[t]$. Frantzikinakis's…

Let $\mathcal{H}$ be a Hardy field, let $G$ be a connected and simply connected nilpotent Lie group, let $\Gamma$ be a uniform discrete subgroup of $G$, and let $X^k=G^k/\Gamma^k$.…

Intermediate value conjecture. There is an element $\phi$ in a Hardy field extension of $H$ such that

The paper considers germs at $+\infty$ of almost regular functions and the structure generated by these germs over the real field. O-minimality conjecture for almost regular germs.…

Let $\mathbb{T}^{\operatorname{da}}$ denote the field of accelero-summable transseries, and let $H$ be a Hardy field. In real accelero-summation, use the organic average whenever s…

Let $H$ be a Hardy field. A series derivation of Hardy type on a generalized series field is a derivation satisfying the series-derivation and Hardy-type conditions described in th…

Let $LE$ denote the Hardy field under discussion, and consider the function $(\log x)(\log\log x)$ for sufficiently large $x$. Hardy's conjecture. The compositional inverse of…

L. van den Dries's conjecture. The structure $\mathbb{R}_{\rm an,powers}$ is a maximal polynomially bounded reduct of $\mathbb{R}_{\rm an,exp}$.

Let $a_1(t),\ldots,a_{\ell}(t)$ be functions belonging to the same Hardy field and having polynomial growth. Suppose that every non-trivial linear combination $a(t)$ of these funct…

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let “the growth assumptions of Theorem A (or Conjecture A)” refer to the hyp…

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let $\bar d(\Lambda)$ denote the upper density of $\Lambda\subset\mathbb{Z}$…