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

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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.

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Equivalent formulations 1

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).

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Primary source

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

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Progress summary

Equivalent formulations 1

The almost-everywhere Weyl-sum exponent conjecture

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