New Bound for Roth’s Theorem
with Generalized Coefficients

The orientation-preserving similitudes of the plane are exactly the transformations of the form $z\mapsto uz+v$ with $u,v\in\mathbf{C}$, $u\neq 0$. Let $p_{1},p_{2}$ and $p_{3}$ be the (distinct) vertices of $T$.

Finding triangles in $A$ that are directly similar to $T$ is equivalent to solving the system of equations

$$\left\{\begin{array}[]{c}up_{1}+v=a_{1}\\ up_{2}+v=a_{2}\\ up_{3}+v=a_{3}\end{array}\right.$$

for $u\in\mathbf{C}\setminus\{0\}$, $v\in\mathbf{C}$ and $a_{1},a_{2},a_{3}\in A$. This system is equivalent to the single equation

$$(p_{3}-p_{2})a_{1}+(p_{1}-p_{3})a_{2}+(p_{2}-p_{1})a_{3}=0,$$

to be solved for distinct $a_{1},a_{2},a_{3}\in A$.

Define $M_{1}$, $M_{2}$ and $M_{3}$ to be the matrices corresponding to multiplication by $p_{3}-p_{2}$, $p_{1}-p_{3}$ and $p_{2}-p_{1}$ in the $\mathbf{Z}$-basis $(\omega_{1},\omega_{2})$ of $\Lambda$. These matrices sum to zero, are nonsingular as the $p_{i}$’s are distinct, and have integer coefficients since $p_{i}\omega_{j}\in\Lambda$ for all $i,j$. We conclude by Corollary 1.3. ∎

Let $A$ be a subset of $\mathbf{Z}^{d}$ containing only finitely many non-trivial solutions to the equation

$$M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0.$$

(2)

We want to prove that

$$\sum_{a\in A\setminus\{0\}}\frac{1}{\left\|a\right\|^{d}}<+\infty.$$

(3)

After removing a finite number of elements from $A$, we can assume that $A$ has no non-trivial solution to Eq. 2.

For $T\geq 1$, let $A_{T}$ be the truncated set

$$A_{T}:=A\cap[-T,T]^{d}.$$

It is sufficient to prove that, for all $T\geq 3$,

$$|A_{T}|\ll\frac{T^{d}}{(\log{T})^{1+c}},$$

(4)

where $c>0$ is the constant from Theorem 1.1.⁴⁴4In this proof, the implied constants in the asymptotic notation $\ll$ and $\asymp$ depend only on the dimension $d$ and the matrices $M_{1},M_{2},M_{3}$. Indeed, we have

$$\sum_{\begin{subarray}{c}a\in A\setminus\{0\}\\ \left\|a\right\|_{\infty}\leq M\end{subarray}}\frac{1}{\left\|a\right\|^{d}}\asymp\sum_{N=1}^{M}\frac{1}{N^{d}}\cdot\#\{a\in A:\left\|a\right\|_{\infty}=N\},$$

and by partial summation, together with Eq. 4, we get

$$\sum_{\begin{subarray}{c}a\in A\setminus\{0\}\\ \left\|a\right\|_{\infty}\leq M\end{subarray}}\frac{1}{\left\|a\right\|^{d}}\asymp\frac{|A_{M}|}{M^{d}}+\int_{1}^{M}\frac{|A_{T}|}{T^{d+1}}\,\mathrm{d}T\ll 1+\int_{2}^{M}\frac{1}{T(\log{T})^{1+c}}\,\mathrm{d}T\ll 1.$$

Taking $M\to+\infty$ proves Eq. 3.

Let $T\geq 3$. Let

$$C=\max\big{(}\left\|M_{1}\right\|_{\mathrm{op}},\left\|M_{2}\right\|_{\mathrm{op}},\left\|M_{3}\right\|_{\mathrm{op}},\lvert\det M_{1}\rvert,\lvert\det M_{2}\rvert,\lvert\det M_{3}\rvert\big{)},$$

where $\left\|M_{i}\right\|_{\mathrm{op}}$ is the operator norm of the matrix $M_{i}$, viewed as a map $(\mathbf{R}^{d},\left\|\cdot\right\|_{\infty})\to(\mathbf{R}^{d},\left\|\cdot\right\|_{\infty})$. Let $p$ be a prime number between $4CT$ and $8CT$, which exists by Bertrand’s postulate.

We embed $A_{T}$ in the abelian group $(\mathbf{Z}/p\mathbf{Z})^{d}$. Let $\overline{A_{T}}$, $\overline{M_{1}},\overline{M_{2}}$ and $\overline{M_{3}}$ be the reductions of $A_{T},M_{1},M_{2}$ and $M_{3}$ modulo $p$. Clearly, each $\overline{M_{i}}$ is invertible as its determinant is not divisible by $p$.

We claim that the map

$$\{(a_{1},a_{2},a_{3})\in(A_{T})^{3}:M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0\}\to\{(x_{1},x_{2},x_{3})\in(\overline{A_{T}})^{3}:\overline{M_{1}}x_{1}+\overline{M_{2}}x_{2}+\overline{M_{3}}x_{3}=0\}$$

given by reduction modulo $p$ is surjective. Indeed, if $(a_{1},a_{2},a_{3})\in(A_{T})^{3}$ is such that

$$M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}\equiv 0\pmod{p},$$

then $M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}=0$ in $\mathbf{R}^{d}$ since we also have

$$\left\|M_{1}a_{1}+M_{2}a_{2}+M_{3}a_{3}\right\|_{\infty}\leq 3CT<p.$$

It follows that $\overline{A_{T}}$ only has trivial solutions to the equation $\overline{M_{1}}x_{1}+\overline{M_{2}}x_{2}+\overline{M_{3}}x_{3}=0$. By Theorem 1.1, we obtain

$$|A_{T}|=|\overline{A_{T}}|\ll\frac{p^{d}}{(\log p^{d})^{1+c}}\asymp\frac{T^{d}}{(\log T)^{1+c}},$$

which proves Eq. 4 and concludes the proof of Corollary 1.3. ∎

This follows directly from [1, Corollary 3.7], applied to the sets $A_{1}$, $T_{2}A_{2}$ and $-T_{3}A_{3}$. Note that, since $B$ is a regular Bohr set,

$$|(B+B^{\prime})\setminus B|\leq|B_{1+\rho}\setminus B|\leq(1+O(d\rho))|B|,$$

so that $B^{\prime}$ is $(2^{-270}\alpha)$-sheltered by $B$, provided that $c$ is sufficiently small (see [1] for the definition of ‘sheltered’ in this context). We see in a similar way that $B^{\prime\prime}$ has the required amount of shelter. Finally, if $x\in B^{\prime\prime}$ and $\gamma^{\prime}\in\langle\widetilde{\Gamma}\rangle$, say $\gamma^{\prime}=\sum_{\gamma\in\Gamma}\varepsilon_{\gamma}\gamma$ with $\varepsilon_{\gamma}\in\{-1,0,1\}$, we have

$$\lvert 1-\gamma^{\prime}(x)\rvert\leq\sum_{\gamma\in\widetilde{\Gamma}}\lvert 1-\gamma(x)^{\varepsilon_{\gamma}}\rvert\leq\tfrac{1}{4}$$

as required. ∎

Let $\varepsilon=c/2$, where $c$ is the constant of Lemma 4.2. We apply Lemma 3.8 with $B_{1}=(B^{\star})_{\rho}$, $B_{2}=T_{2}^{-1}(B^{\star})_{\rho\rho^{\prime}}$, $B_{3}=T_{3}^{-1}(B^{\star})_{\rho}$, where $\rho=c_{1}\alpha\varepsilon/d$ and $\rho^{\prime}=c_{2}\alpha/d$, with $c_{1},c_{2}>0$ being two small constants, chosen in particular such that $B_{1}$, $B_{2}$ and $B_{3}$ are regular.

If the second case of Lemma 3.8 holds, then $\mathcal{A}$ has a density increment of strength $[1,0;O(1)]$ relative to one of the $B_{i}$’s. By Lemma 3.7, this implies that $\mathcal{A}$ has a density increment of strength $[1,0;\tilde{O}_{\alpha}(1)]$ relative to $B^{\star}$, $T_{2}^{-1}B^{\star}$ or $T_{3}^{-1}B^{\star}$.

We may thus suppose that the first case of Lemma 3.8 holds. That is, there is some $x\in G$ such that, if we let

$$A_{1}=(\mathcal{A}-x)\cap B_{1},\quad A_{2}=(\mathcal{A}-x)\cap B_{2}\quad\text{and}\quad A_{3}=(\mathcal{A}-x)\cap B_{3},$$

then each $A_{i}$ has relative density at least $(1-\varepsilon)\alpha$ in the corresponding $B_{i}$. We now use Lemma 4.2 with $B=(B^{\star})_{\rho}$ and $B^{\prime}=(B^{\star})_{\rho\rho^{\prime}}$.

1. 1.

   In the first case, we get

   $$T(A_{1},A_{2},A_{3})\gg(1-\varepsilon)^{3}\alpha^{3}\mu(B)\mu(B^{\prime})\gg\exp\left(-\tilde{O}_{\alpha}(d\log(2d)\right)\mu(B^{\star})^{2},$$

   where the last inequality follows from Lemma 3.3. Since the $A_{i}$’s are subsets of the same translate of $\mathcal{A}$ and the equation $a_{1}+T_{2}a_{2}+T_{3}a_{3}=0$ is translation-invariant, we have

   $${T(\mathcal{A},\mathcal{A},\mathcal{A})\geq T(A_{1},A_{2},A_{3})},$$

   which gives the claimed bound.

2. 2.

   In the second case, there is a regular Bohr set $B^{\prime\prime}$ as in the statement of the lemma such that

   $$\left\|1_{A}*\mu_{B^{\prime\prime}}\right\|_{\infty}\geq(1+c)(1-\varepsilon)\alpha\geq(1+c/4)\alpha,$$

   (7)

   where $A$ is either $A_{1}$ or $-T_{3}A_{3}$. We therefore deduce that $\mathcal{A}$ has a density increment of strength $[1,\alpha^{-1};\tilde{O}_{\alpha}(1)]$ relative to $(B^{\star})_{\rho\rho^{\prime}}$ or $T_{3}^{-1}(B^{\star})_{\rho\rho^{\prime}}$. By Lemma 3.7, this means that $\mathcal{A}$ has a density increment of the same strength relative to $B^{\star}$ or $T_{3}^{-1}B^{\star}$.∎

Let $C=\tilde{O}_{\alpha}(1)$ be the constant in the density increment case of Proposition 4.3 ($C$ is fixed as $\alpha$ is given). Recall that $\tilde{O}_{\alpha^{\prime}}(1)$ is short for $C_{1}\log(2/\alpha^{\prime})^{C_{2}}$, which is a decreasing function of $\alpha^{\prime}$. Thus, if we use Proposition 4.3 with some pair $\mathcal{A}^{\prime}\subset\mathcal{B}^{\prime}$ having relative density $\alpha^{\prime}\geq\alpha$ and the second case applies, we will have a density increment of strength $[1,(\alpha^{\prime})^{-1};C]$.

We inductively construct two sequences $(A_{n})$ and $(B_{n})$, where, for each $n$, $A_{n}$ is a subset of $B_{n}$ with relative density $\alpha_{n}$. Let $A_{0}=A$ and $B_{0}=B$. Assume that $A_{i}$ and $B_{i}$ have been constructed for $i<n$. We use Proposition 4.3 with $\mathcal{A}=A_{n-1}$ and $\mathcal{B}=B_{n-1}$. If the first case of the proposition holds, we stop the construction. Otherwise, we are in the density increment case and there are sets $A_{n}\subset B_{n}$ such that

• •

  $A_{n}$ is a subset of a translate of $A_{n-1}$;

• •

  $A_{n}$ is a subset of $B_{n}$ of relative density $\alpha_{n}\geq(1+C^{-1})\alpha_{n-1}$;

• •

  $B_{n}$ is a regular Bohr set of the form

  $$B_{n}=(S_{n}B_{n-1}^{\star})_{\rho_{n}}\cap\widetilde{B}_{n},$$

  (8)

  where

  • –

    $B_{n-1}^{\star}$ is the Bohr set

    $$B_{n-1}^{\star}:=B_{n-1}\cap T_{2}B_{n-1}\cap T_{3}B_{n-1},$$

    (9)

    whose rank we denote by $d_{n-1}^{\star}$,

  • –

    $S_{n}$ is either $\mathrm{Id}_{G}$, $T_{2}^{-1}$ or $T_{3}^{-1}$,

  • –

    $\widetilde{B}_{n}=\mathrm{Bohr}(\widetilde{\Gamma}_{n},\widetilde{\nu}_{n})$, where $|\widetilde{\Gamma}_{n}|\leq C\alpha^{-1}$ and

    $$\bigg{(}\frac{\rho_{n}}{4}\bigg{)}^{d_{n-1}^{\star}}\prod_{\gamma\in\widetilde{\Gamma}_{n}}\frac{\widetilde{\nu}_{n}(\gamma)}{8}\geq\exp\left(-\tilde{O}_{\alpha}\left(d_{n-1}^{\star}+\alpha^{-1}\right)\log\left(d_{n-1}^{\star}\right)\right).$$

    (10)