(a) [3 marks] Let $q$ be a prime, and let $a, b$ be integers, not both congruent to 0 modulo $q$. Show that $$ \left|\sum_{1 \leqslant n \leqslant q} e\left(\frac{a n^{2}+b n}{q}\right)\right| \ll q^{1 / 2} $$ [Hint: Complete the square and apply a bound for the Gauss sums.] (b) [ $[5$ marks Still letting $q$ be a prime and $a \not \equiv 0(\bmod q)$, show that $$ \left|\sum_{1 \leqslant n \leqslant q} e\left(\frac{a\left(n^{3}+n\right)}{q}\right)\right| \ll q^{3 / 1} $$ [Hint: Adapt proofs from the course and use part (a). (c) [7 marks] Show that if $q$ is a large enough prime, then for every $N \in \mathbb{Z}$ there is a solution $\left(n_{1}, n_{2}, \ldots, n_{5}\right)$ to the congruence $$ \left(n_{1}^{3}+n_{1}\right)+\left(n_{2}^{3}+n_{2}\right)+\cdots+\left(n_{5}^{3}+n_{5}\right) \equiv N \quad(\bmod q) $$ [Hint: Write a formula for the number of solutions in terms of Fourier transforms and use part (b).] (d) [10 marks] Show that the conclusion of part (c) continues to hold if the primality condition on $q$ is replaced by the condition $(q, m)=1$ for some suitable fixed integer $m \geqslant 1$. [Hint: You may lift the result of part (c) to prime power moduli by proving the following. If $p$ is odd and $a$ is an integer with $a \neq 0(\bmod p), 27 a^{2}+2 \not \equiv 0(\bmod p)$, then the solvability of $x^{3}+x \equiv a(\bmod p)$ implies the solvability of $x^{3}+x \equiv a\left(\bmod p^{\ell}\right)$ for every $\ell \geqslant 1 .]$
Proof .
Problem 2.
(a) [2 marks] Let $A \subset[N]$. Give a formula for the number of solutions to $$ 2 x+3 y=5 z, \quad x, y, z \in A $$ involving the Fourier transform of the set $A$. (b) $[4$ marks $]$ Let $A \subset[N],|A|=\delta N, \delta>N^{-1 / 10} .$ Show that if all solutions to (1) have $x=y=z$, then $$ \sup {\alpha \in \mathbb{R}}\left|\widehat{f{A}}(\alpha)\right| \gg \delta^{2} N $$ where $f_{A}=1_{A}-\delta 1_{[N]}$. (c) $[7$ marks $]$ Let $A \subset[N]$ satisfy $|A|=\delta N$ with $N \geqslant N_{0}(\delta)$ large enough. Sketch a proof that (1) has a solution with $x, y, z$ not all equal. [You should give the basic structure of the argument, but need not supply full details. (d) [1 mark] Show that if $N$ is large enough, there exists a set $A \subset[N]$ of size $|A|>N / 3$ such that $A$ contains no solutions to $$ 2 x+3 y=4 z, \quad x, y, z \in A $$ (e) [11 marks] Consider the set $$ A=\left{n \leqslant N:|\sqrt{2} n-1 / 3|_{\mathbb{R} / \mathbb{Z}} \leqslant \frac{1}{100}\right} $$ (i) Show that $A$ contains no solutions to $(2)$. (ii) Show that if $N$ is large enough in terms of $\varepsilon$, then $A$ is ” $\varepsilon$ -uniform” in the following sense: for every arithmetic progression $P \subset[N]$ with $|P| \geqslant \varepsilon N$ we have $$ || P \cap A\left|-\frac{|P||A|}{N}\right| \leqslant \varepsilon|P| $$ [You may use the following fact: for any $\delta, \eta>0$ there exist trigonometric polynomials $T^{\pm}(x)=\sum_{|m| \leqslant M} c_{m}^{\pm} e^{i m x}$ (with $M, c_{m}^{\pm}$ depending on $\delta, \eta$ ) such that $$ T^{-}(x) \leqslant 1_{|x|_{\mathbb{R} / \mathrm{z}} \leqslant \delta / 2} \leqslant T^{+}(x) $$ and $\left.c_{0}^{-}=\delta(1-\eta), c_{0}^{+}=\delta(1+\eta) .\right]$
