Let d be the degree parameter, let mathsfT_d be the associated torus, and for mathbfx∈mathsfT_d define sigma(mathbfx)=limsup_N rightarrow∞fraclog |S_d(mathbfx;N)|log N. For each 0 leqslantalpha leqslant1, let Omega_alpha=mathbfx∈mathsfT_d:sigma(mathbfx)=alpha. Almost-everywhere Weyl-sum exponent conjecture. For alpha∈[0,1], lambda(Omega_alpha)= 0&for alpha≠ 1/2,1&for alpha=1/2. Thus, almost every parameter has Weyl-sum exponent 1/2, while every other level set has measure zero. The statement is presented as a stronger conjecture than the preceding claim that lambda(Omega_alpha)=0 for alpha∈(1/2,1]; its resolution is not specified in the supplied text.

Almost-everywhere Weyl-sum exponent conjecture

About 7 years old · traced to 2019

Let $d$ be the degree parameter, let $\mathsf{T}_d$ be the associated torus, and for $\mathbf{x}\in\mathsf{T}_d$ define

\sigma(\mathbf{x})=\limsup_{N\rightarrow\infty}\frac{\log |S_d(\mathbf{x};N)|}{\log N}.

For each $0\leqslant\alpha\leqslant1$ , let

\Omega_\alpha=\{\mathbf{x}\in\mathsf{T}_d:\sigma(\mathbf{x})=\alpha\}.

Almost-everywhere Weyl-sum exponent conjecture. For $\alpha\in[0,1]$ ,

\lambda(\Omega_\alpha)=\begin{cases}0&\text{for }\alpha\neq 1/2,\\1&\text{for }\alpha=1/2. \end{cases}

Thus, almost every parameter has Weyl-sum exponent $1/2$ , while every other level set has measure zero. The statement is presented as a stronger conjecture than the preceding claim that $\lambda(\Omega_\alpha)=0$ for $\alpha\in(1/2,1]$ ; its resolution is not specified in the supplied text.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The almost-everywhere Weyl-sum exponent conjecture
For an integer $d\geqslant 2$ , let $\mathsf{T}_d=({\mathbb R}/{\mathbb Z})^d$ and define
$S_d(\mathbf x;N)=\sum_{n=1}^{N}{\mathbf{\,e}}(x_1n+\ldots+x_dn^d),$
where ${\mathbf{\,e}}(x)=\exp(2\pi i x)$ . For $0<\alpha<1$ , let
${\mathcal E}_{\alpha,d}=\{\mathbf x\in\mathsf{T}_d:|S_d(\mathbf x;N)|\geqslant N^\alpha\text{ for infinitely many }N\in{\mathbb N}\},$
and set
$\vartheta_d=\inf\{\alpha>0:\lambda({\mathcal E}_{\alpha,d})=0\}.$
The almost-everywhere Weyl-sum exponent conjecture. For each integer $d\geqslant 2$ we have
$\vartheta_d=1/2.$
The Menshov--Rademacher theorem gives the upper bound $\vartheta_d\leqslant 1/2$ ; the conjecture asserts that this bound is sharp for every degree.
source: Changhao Chen and Igor E. Shparlinski, “On Large Values of Weyl Sums”, arXiv:1901.01551 (2020).

References

Primary source

Changhao Chen and Igor E. Shparlinski, “On Large Values of Weyl Sums”, arXiv:1901.01551 (2020).

Posted 1 month ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.

The almost-everywhere Weyl-sum exponent conjecture

Primary source

Progress summary