Polynomial degree dependence and explicit k-dependence in simultaneous small fractional parts of polynomials

Maynard (Geom. Funct. Anal. 2021) proved that for any $k$ real polynomials $f_1,\ldots,f_k$ of degree at most $d$, all vanishing at $0$, there is an integer $n<x$ such that the distance of every $f_i(n)$ to the nearest integer is at most $x^{-c_d/k}$, with a constant $c_d>0$ depending only on the degree $d$; the $1/k$ shape in the number of polynomials is essentially optimal, but the proof only gives $c_d$ exponentially small in $d$ (about $10^{-d}$), and the dependence of the implied constant on $k$ and $d$ is not tracked. Maynard explicitly left open the quantitative refinement: he suggested that with modest additional effort $c_d$ should be of order $C/d^2$, or even the hybrid $C/(d+d^2/k)$, and that the implied constants should be makeable explicit in $k$ and $d$. A 2024 preprint of Lau improved the $d$-aspect to $c_d$ of order $1/(10.5\,d^2)$, but it is unpublished, and it leaves the hybrid exponent form and the explicit $k$-dependence of the constants open. The problem posed here is to establish the simultaneous small fractional parts bound in fully explicit, near-optimal quantitative form: with the exponent constant of polynomial (ideally hybrid $C/(d+d^2/k)$) size in the degree, and with an implied constant that is explicit as a function of $k$ and $d$.

Background

For a real number $t$, write $\|t\|_{\mathbb{R}/\mathbb{Z}}$ for the distance from $t$ to the nearest integer. A classical problem of analytic number theory asks how small the fractional parts of a polynomial can be made along the integers: given $f\in\mathbb{R}[X]$ with $f(0)=0$ (a necessary hypothesis, since $f(n)=n+1/2$ never approaches an integer), one wants a bound on $\min_{1\le n\le x}\|f(n)\|_{\mathbb{R}/\mathbb{Z}}$ that is uniform over all choices of the coefficients of $f$. Writing $d$ for the degree, the current single-polynomial record is due to Baker: $\min_{n\le x}\|f(n)\|_{\mathbb{R}/\mathbb{Z}}\ll_d x^{-1/(2d(d-1))+o(1)}$ for $d\ge 8$. The exponent rests on the modern technology for Vinogradov's mean value theorem (VMVT), i.e. the sharp mean-value estimates for Weyl exponential sums $\sum_{n\le N} e(\alpha_1 n+\cdots+\alpha_d n^d)$ with $e(t)=e^{2\pi i t}$: first Wooley's efficient congruencing method and then the proof of the VMVT main conjecture by Bourgain, Demeter and Guth. This record is believed to be far from optimal: a conjecture of R. C. Baker predicts that the exponent should be improvable to $1+o(1)$, and no feasible approach to exponents of that shape is known.

The simultaneous version asks for one integer $n$ that makes all $k$ fractional parts small at once: for polynomials $f_1,\ldots,f_k\in\mathbb{R}[X]$ of degree at most $d$ with $f_1(0)=\cdots=f_k(0)=0$, bound $\min_{1\le n\le x}\max_{1\le i\le k}\|f_i(n)\|_{\mathbb{R}/\mathbb{Z}}$, again uniformly over all coefficient choices. Schmidt's 1977 monograph (which introduced the 'determinant argument', an increment strategy in which one either finds equidistribution at the desired scale or extracts an additive-structure obstruction and passes to a self-similar subproblem with fewer polynomials) gave exponents of order $1/k^2$ for $d=2$; Baker and Harman obtained the exponent $1/(k^2+k^{c_d})$; and choosing the coefficients at random shows that a bound uniform over all systems cannot beat order $x^{-1/k}$. Maynard (2021) reached the optimal shape in the $k$-aspect: for some $c_d>0$ depending only on $d$, one has $\min_{n\le x}\max_{1\le i\le k}\|f_i(n)\|_{\mathbb{R}/\mathbb{Z}}\ll_{d,k}x^{-c_d/k}$. In the terminology of Baker's 1979 survey, this confirms that every such system has 'Heilbronn status': the simultaneous fractional parts become arbitrarily small as $x\to\infty$ for every coefficient choice, the general-polynomial analogue of Heilbronn's classical theorem that $\|\alpha n^2\|_{\mathbb{R}/\mathbb{Z}}$ becomes arbitrarily small for every real $\alpha$.

While the $1/k$ shape is essentially optimal, the quantitative form of Maynard's result is not. As presented, the constant $c_d$ in the exponent $c_d/k$ is exponentially small in $d$: Maynard notes that $c_d=10^{-d}$ would probably suffice for his argument. He states explicitly that with only a small amount of additional effort the constant could probably be taken of the form $c_d=C/d^2$, or perhaps even the hybrid $c_d=C/(d+d^2/k)$, where the exponent constant is now allowed to depend on $k$ as well as on $d$ (which is stronger than the uniform $C/d^2$ by a factor of order $d$ when $k\gg d$), for a relatively small explicit absolute constant $C$; and that, although his paper makes no effort to control it, the implied constant's dependence on $k$ and $d$ should be recoverable by adapting the ideas behind Proposition A.2 of Green and Tao's work on Szemerédi's theorem (a simultaneous quadratic recurrence statement whose proof follows Schmidt's argument and admits explicit constants). In June 2024, Lau, following Maynard's argument with more careful choices of constants and with a new multidimensional 'same denominators' lemma (a multidimensional form of Maynard's Lemma 7.3, replacing $d$ iterations of a one-dimensional argument), obtained the claimed improvement to the $C/d^2$ form: $\min_{n\le x}\max_{1\le i\le k}\|f_i(n)\|_{\mathbb{R}/\mathbb{Z}}\ll_{d,k,\varepsilon}x^{-1/((10.5+\varepsilon)kd(d-1))}$. This preprint leaves open the hybrid form of the exponent, the explicit dependence of the implied constants on $k$ and $d$, and the size of the absolute constant; and, being unrefereed, it constitutes strong claimed progress on, but not settled closure of, the $d$-aspect.

Problem Statement

Let $k,d\ge 2$ and let $f_1,\ldots,f_k\in\mathbb{R}[X]$ be real polynomials of degree at most $d$ with $f_1(0)=\cdots=f_k(0)=0$, with all bounds required to be uniform over the coefficients. Maynard (2021) proved that for some $c_d>0$ depending only on $d$ there is a positive integer $n<x$ with $\|f_i(n)\|_{\mathbb{R}/\mathbb{Z}}\ll_{d,k}x^{-c_d/k}$ simultaneously for all $1\le i\le k$, which is essentially optimal in the $k$-aspect; but as presented $c_d$ is exponentially small in $d$ and the implied constant's dependence on $k$ and $d$ is not explicit. The problem is to establish a quantitative form of this theorem with:

(1) Degree aspect: an exponent constant of polynomial size in $1/d$ - specifically, prove the bound with $c_d\ge C/d^2$ for an explicit absolute constant $C$ (currently claimed only in the unpublished 2024 preprint of Lau, with $c_d=1/((10.5+\varepsilon)d(d-1))$), and ideally the hybrid form, in which the exponent constant is allowed to depend on $k$ as well as on $d$, i.e. $\min_{1\le n\le x}\ \max_{1\le i\le k}\|f_i(n)\|_{\mathbb{R}/\mathbb{Z}}\ \ll\ x^{-C/(kd+d^2)},$ valid for all sufficiently large $x$ with the implied constant required to be explicit in $k$ and $d$ (component (2)); this is stronger than the uniform $C/d^2$ form by a factor of order $d$ when $k\gg d$.

(2) Explicit constants: a version of the bound in which the implied constant is explicit as a function of $k$ and $d$ (for example, of polynomial growth).

Component (3), reducing the absolute constant (10.5 in Lau's preprint, versus 2 in the single-polynomial analogue $1/(2d(d-1))$), is a further quality dimension; the conjectural optimum, suggested by the random-coefficient heuristic, is the exponent $1/k$ with a $d$-independent constant, tied to Baker's long-open single-polynomial conjecture of an exponent $1+o(1)$ and not expected to be reachable by current techniques. Components (1) and (2) serve one objective, a fully explicit and near-optimal quantitative form of Maynard's simultaneous small fractional parts theorem in all parameters $x$, $k$, and $d$.

References

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