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This question is a close cousin of the following:

Lower bound on exponential sums

Let $\phi:[0,1]\to \mathbb R$ be a smooth function with $\frac 1 {10}<\phi'< 10, \frac 1 {10}<\phi''< 10$. Consider the exponential sum $$ f(x,y)=\sum_{n=1}^N \exp(2\pi i (x N^{-1}n+y\phi(N^{-1}n))). $$ Is it true that $$ \left\|f\right\|_{L^6([0,1]^2)}\gtrsim N^{\frac 1 2}\log N? $$ It is known that when $\phi(n)=n^2$, this can be proved by an elementary number-theoretic method. But is it true for a general $\phi$?

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    $\begingroup$ I want to interpret $\phi'\sim1$ to mean $\phi'$ is asymptotic to one, but since $\phi$ is only defined on a finite interval, that can't be right, can it? But then, the summation doesn't make sense if $\phi$ is only defined on $[0,1]$, since it involves $\phi(n)$ for $n=1,2,\dots,N$. So, what do you really mean? $\endgroup$
    – Gerry Myerson
    Commented Nov 6, 2020 at 11:51
  • $\begingroup$ @GerryMyerson Sorry for the confusion. By a function $\phi\sim1$ I meant that $\phi$ is both bounded above and below by absolute constants. $\endgroup$
    – Thomas Yang
    Commented Nov 6, 2020 at 22:06
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    $\begingroup$ OK. But you still have $\phi$ defined on $[0,1]$ but evaluated at $2,3,4,\dots,N$. Something's gotta give. $\endgroup$
    – Gerry Myerson
    Commented Nov 6, 2020 at 22:32
  • $\begingroup$ @GerryMyerson Ah that is my bad. I forgot to multiply $N^{-1}$. $\endgroup$
    – Thomas Yang
    Commented Nov 7, 2020 at 0:40

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