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David,

At first sight I think you might be right about this. Personally, I try to avoid using Weyl's inequality in this form, but rather some statement of the following form: If $|S_N| \geq \epsilon N$, and if $\epsilon > N^{-c}$, then there is some $q \leq \epsilon^{-C}$ such that the fractional part of $q\theta$ is at most $\epsilon^{-C}/N^d$.

In other words: if the exponential sum is large then $\theta$ is very close to a rational with denominator $q$.

You can prove

I sketch a proof of this log-free'' variant in much the same way, that is to say by repeated squaringbelow. I don't think the inequality is typically stated in this way because so far as I'm aware it's more effort to prove, and because the form you stated leads to better bounds in is just fine for Waring's problem, where the factor of $(\log N)^C$ isn't very important. However, as you point out, it does seem to be important when talking about the equidistribution result (although normally one wouldn't involve quantitative estimates when talking about equidistribution results of the type you state).

Let me try to be a little more specific about how to prove this log free'' variant of Weyl's inequality that I've mentioned. Presumably there is a reference in the literature. However you can start from the presentation that I give on pages 59-60 of these notes

At some point one obtains many $h_1, h_2, \dots, h_d$ for which $\Vert \theta h_1,\dots, h_d \Vert$ is small. At this point it is standard to invoke the divisor function estimate to show that there are in fact many $n$ for which $\Vert \theta n \Vert$ is small. However in doing this one loses an $N^{\epsilon}$ (it's worse than $\log^C N$ - are you sure you've quoted Gowers accurately?). To avoid losing it, let $S$ be the set of all $h_1\dots h_d$ mentioned above. Then $\Vert \theta (s_1 + s_2 + \dots + s_m) \Vert$ is small for all choices of $s_1,\dots, s_m \in S$, and one can argue* that for big enough $m$ this set of sums of $S$ is really big (i.e. there is no loss of $N^{\epsilon}$.)

*The key point is that for large enough $m$, the number of representations of any $n \in [X^d, 2X^d]$ as a sum of $m$ things of the form $h_1 \dots h_d$, $h_i \sim X$ is bounded by $C X^{d(m-1)}$. The usual proof would have $m = 1$, where this statement is actually false. The problem is, I think I'd need to use the Hardy-Littlewood method (which uses Weyl's inequality, but only the weaker form with the $N^{\epsilon}$) to prove this statement! Little surprise that you don't find this argument in textbooks then.

Actually, I'd be very interested to see a decent reference for all this.

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David,

At first sight I think you might be right about this. Personally, I try to avoid using Weyl's inequality in this form, but rather some statement of the following form: If $|S_N| \geq \epsilon N$, and if $\epsilon > N^{-c}$, then there is some $q \leq \epsilon^{-C}$ such that the fractional part of $q\theta$ is at most $\epsilon^{-C}/N^d$.

In other words: if the exponential sum is large then $\theta$ is very close to a rational with denominator $q$.

You can prove this variant in much the same way, that is to say by repeated squaring. I don't think the inequality is typically stated in this way because the form you stated leads to better bounds in Waring's problem, where the factor of $(\log N)^C$ isn't very important. However, as you point out, it does seem to be important when talking about the equidistribution result.

Let me try to be a little more specific about how to prove this log free'' variant of Weyl's inequality that I've mentioned. Presumably there is a reference in the literature. However you can start from the presentation that I give on pages 59-60 of these notes

At some point one obtains many $h_1, h_2, \dots, h_d$ for which $\Vert \theta h_1,\dots, h_d \Vert$ is small. At this point it is standard to invoke the divisor function estimate to show that there are in fact many $n$ for which $\Vert \theta n \Vert$ is small. However in doing this one loses an $N^{\epsilon}$ (it's worse that than $\log^C N$ - are you sure you've quoted Gowers accurately?). To avoid losing it, let $S$ be the set of all $h_1\dots h_d$ mentioned above. Then $\Vert \theta (s_1 + s_2 + \dots + s_m) \Vert$ is small for all choices of $s_1,\dots, s_m \in S$, and one can argue* that for big enough $m$ this set of sums of $S$ is really big (i.e. there is no loss of $N^{\epsilon}$.)

*The key point is that for large enough $m$, the number of representations of any $n \in [X^d, 2X^d]$ as a sum of $m$ things of the form $h_1 \dots h_d$, $h_i \sim X$ is bounded by $C X^{d(m-1)}$. The usual proof would have $m = 1$, where this statement is actually false. The problem is, I think I'd need to use the Hardy-Littlewood method (which uses Weyl's inequality, but only the weaker form with the $N^{\epsilon}$) to prove this statement! Little surprise that you don't find this argument in textbooks then.

Actually, I'd be very interested to see a decent reference for all this.

Let me try to be a little more specific about how to prove this log free'' variant of Weyl's inequality that I've mentioned. Presumably there is a reference in the literature. However you can start from the presentation that I give on pages 59-60 of these notes
At some point one obtains many $h_1, h_2, \dots, h_d$ for which $\Vert \theta h_1,\dots, h_d \Vert$ is small. At this point it is standard to invoke the divisor function estimate to show that there are in fact many $n$ for which $\Vert \theta n \Vert$ is small. However in doing this one loses an $N^{\epsilon}$ (it's worse that $\log^C N$ - are you sure you've quoted Gowers accurately?). To avoid losing it, let $S$ be the set of all $h_1\dots h_d$ mentioned above. Then $\Vert \theta (s_1 + s_2 + \dots + s_m) \Vert$ is small for all choices of $s_1,\dots, s_m \in S$, and one can argue that for big enough $m$ this set of sums of $S$ is really big (i.e. there is no loss of $N^{\epsilon}$.)