10 Large value additive energy

10.1 Additive energy

Definition 10.1 Additive energy

Let $W$ be a finite set of real numbers. The additive energy $E_{1} (W)$ of such a set is defined to be the number of quadruples $(t_{1}, t_{2}, t_{3}, t_{4}) \in W$ such that

| t_{1} + t_{2} - t_{3} - t_{4} | \leq 1.

We remark that in additive combinatorics, the variant $E_{0} (W)$ of the additive energy is often studied, in which $t_{1} + t_{2} - t_{3} - t_{4}$ is not merely required to be $1$ -bounded, but in fact vanishes exactly. However, this version of additive energy is less relevant for analytic number theory applications.

Lemma 10.2 Basic properties of additive energy

If $W$ is a finite set of reals, then

$E_{1} (W) ≍ \int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq 1} |^{2} d x .$

More generally, for any $r > 0$ we have

$E_{1} (W) ≍ r^{O (1)} \int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq r} |^{2} d x .$
If $W$ is a finite set of reals, then

$E_{1} (W) ≍ \int_{- 1}^{1} | \sum_{t \in W} e (t θ) |^{4} d θ .$
If $W_{1}, \dots, W_{k}$ are finite sets of reals, then

$E_{1} (W_{1} \cup \dots \cup W_{k})^{1 / 4} ≪ E_{1} (W_{1})^{1 / 4} + \dots + E_{1} (W_{k})^{1 / 4} .$
If $W$ is $1$ -separated and contained in an interval of length $T \geq 1$ , then

$(# W)^{2}, (# W)^{4} / T ≪ E_{1} (W) ≪ (# W)^{3} .$
If $W$ is contained in an interval $I$ , which is in turn split into $K$ equally sized subintervals $J_{1}, \dots, J_{K}$ , then

$E_{1} (W)^{1 / 3} ≪ \sum_{k = 1}^{K} E_{1} (W \cap J_{k})^{1 / 3} .$

Note that the lower bound of $(# W)^{4} / T$ would be expected to be attained if the set $W$ is distributed “randomly” and is reasonably large (of size $≫ \sqrt{T}$ ). So getting upper bounds of the additive energy of similar strength to this lower bound can be viewed as a statement of “pseudorandomness” (or “Gowers uniformity”) of this set.

Proof ▶

For (i), we just prove the first estimate, as the second follows from the first by several applications of the triangle inequality. The right-hand side can be expanded as

\sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W} | {x : | t_{1} + t_{2} - x |, | t_{3} + t_{4} - x | \leq 1} | .

Every quadruple contributing to $E_{1} (W)$ then contributes $≫ 1$ to the right-hand side, giving the upper bound. To get the matching lower bound, note that

\sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W} | {x : | t_{1} + t_{2} - x |, | t_{3} + t_{4} - x | \leq 1 / 2} \leq E_{1} (W)

and hence

E_{1} (W) ≫ \int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq 1 / 2} |^{2} d x .

The upper bound then follows from the triangle inequality.

For (ii), we can upper bound the indicator function of $[- 1, 1]$ by the Fourier transform of a non-negative bump function $φ$ , so that the right-hand side is bounded by

\sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W} φ (t_{1} + t_{2} - t_{3} - t_{4})

which is then bounded by $O (E_{1} (W))$ by choosing the support of $φ$ appropriately. The lower bound is established similarly (using the arguments in (i) to adjust the error tolerance $1$ in the constraint $| t_{1} + t_{2} - x | \leq 1$ as necessary.)

For (iii), first observe we may remove duplicates and assume that the $W_{i}$ are disjoint, then we can use (ii) and the triangle inequality.

For (iv), the first lower bound comes from considering the diagonal case $t_{1} = t_{3}, t_{2} = t_{4}$ and the upper bound comes from observing that once $t_{1}, t_{2}, t_{3}$ are fixed, there are only $O (1)$ choices for $t_{4}$ thanks to the $1$ -separated hypothesis. Finally, observe that

\int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq 1} | d x = (2 # W)^{2}

hence by Cauchy–Schwarz

\int_{R} | # {(t_{1}, t_{2}) \in W : | t_{1} + t_{2} - x | \leq 1} |^{2} d x ≫ (# W)^{2} / T

and the claim follows from (i).

For (v), write $a_{k} := E_{1} (W \cap J_{k})^{1 / 4}$ . Each tuple $(t_{1}, t_{2}, t_{3}, t_{4})$ that contributes to $E_{1} (W)$ is associated to a tuple $J_{k_{1}}, J_{k_{2}}, J_{k_{3}}, J_{k_{4}}$ of intervals with $k_{1} + k_{2} - k_{3} - k_{4} = O (1)$ . By modifying the proof of (ii), the total contribution of such a tuple of intervals is

≪ \int_{R} \prod_{j = 1}^{4} | \sum_{t \in W \cap J_{k_{j}}} e (t θ) | d θ

which by Cauchy–Schwarz is bounded by

≪ a_{k_{1}} a_{k_{2}} a_{k_{3}} a_{k_{4}} .

Thus we see that

E_{1} (W) ≪ \sum_{m = O (1)} a * a * \tilde{a} * \tilde{a} (m)

where ${\tilde{a}}_{k} := a_{- k}$ and $*$ denotes convolution on the integers. By Young’s inequality we then have

E_{1} (W) ≪ ‖ a ‖_{ℓ^{4 / 3}}^{4}

and the claim follows.

We remark that (v) can also be proven using [ 32 , Lemma 4.8, (4.2) ] .

We will also study the following related quantity. Given a set $W$ and a scale $N > 1$ , let $S (N, W)$ denote the double zeta sum

S (N, W) := \sum_{t, t^{'} \in W} {| \sum_{n \in [N, 2 N]} n^{- i (t - t^{'})} |}^{2} .

We caution that this normalization differs from the one in [ 110 ] , where $n^{- 1 / 2 - i (t - t^{'})}$ is used in place of $n^{- i (t - t^{'})}$ . This sum may also be rearranged as

S (N, W) = \sum_{n, m \in [N, 2 N]} | R_{W} (n / m) |^{2}

where $R_{W}$ is the exponential sum

R_{W} (x) := \sum_{t \in W} x^{i t} .

From the first formula it is clear that $S (N, W)$ is monotone non-decreasing in $W$ , and from the second formula one has the triangle inequality

S {(N, ⋃_{i = 1}^{k} W_{i})}^{1 / 2} \leq \sum_{i = 1}^{k} S (N, W_{i})^{1 / 2}

when the $W_{i}$ are disjoint, and hence also when they are not assumed to be disjoint, thanks to the monotonicity.

The following Cauchy–Schwarz inequality is also useful:

Lemma 10.3 Cauchy–Schwarz and double

ζ

-sums

[ 16 , Lemma 3.4 ] If $W, W^{'}$ are finite sets of reals, $N > 1$ , and $a_{n}$ is a $1$ -bounded sequence for $n \in [N, 2 N]$ , then

\sum_{t \in W, t^{'} \in W^{'}} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i t} |}^{2} \leq S (N, W)^{1 / 2} S (N, W^{'})^{1 / 2} .

In particular

\sum_{t \in W} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i t} |}^{2} \leq S (N, W) .

Proof ▶

The left-hand side of 4 can be rewritten as

\sum_{n, m \in [N, 2 N]} a_{n} \overset{―}{a_{m}} \overset{―}{R_{W}} (n / m) R_{W^{'}} (n / m) .

The claim is now immediate from 2 and the Cauchy–Schwarz inequality.

To relate $S (N, W)$ to $E_{1} (W)$ , we first observe the following lemma, implicit in [ 76 ] and made more explicit in [ 61 , Lemma 11.4 ] .

Lemma 10.4 Energy controlled by third moment

Suppose that $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is a large value pattern with $T \geq 1$ and $1 \leq N ≪ T^{O (1)}$ . Then

V^{2} E_{1} (W) ≪ T^{o (1)} \sum_{n, m \in [N, 2 N]} | R_{W} (n / m) |^{3} + T^{- 50} .

Proof ▶

By hypothesis, we have

V^{2} E_{1} (W) \leq \sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W : | t_{1} + t_{2} - t_{3} - t_{4} | \leq 1} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i t_{4}} |}^{2} .

By standard Fourier arguments (see [ 61 , Lemma 11.3 ] ), we can bound

| \sum_{n \in [N, 2 N]} a_{n} n^{- i t_{4}} | ≪ T^{o (1)} \int_{t : | t - t_{4} | \leq T^{o (1)}} | \sum_{n \in [N, 2 N]} a_{n} n^{- i t} | d t + T^{- 100} .

Since each $t_{1}, t_{2}, t_{3}$ generates at most $O (1)$ choices for $t_{4}$ , we conclude that

V^{2} E_{1} (W) ≪ T^{o (1)} \sum_{t_{1}, t_{2}, t_{3} \in W} \int_{s : | s | \leq T^{o (1)}} {| \sum_{n \in [N, 2 N]} a_{n} n^{- i (t_{1} + t_{2} - t_{3} + s)} |}^{2} d s + T^{- 50},

The right-hand side can be rewritten as

T^{o (1)} \sum_{n, m \in [N, 2 N]} a_{n} \overset{―}{a_{m}} (n / m)^{- i s} \overset{―}{R_{W}} (n / m)^{2} R_{W} (n / m) + T^{- 50},

and the claim then follows from the triangle inequality.

Thus, $S (N, W)$ involves a second moment of $R_{W}$ , while the energy $E_{1} (W)$ is related to the third moment. Using the trivial bound $| R_{W} (x) | \leq | W |$ we can then obtain the trivial bound

V^{2} E_{1} (W) ≪ T^{o (1)} | W | S (N, W) + T^{- 50}

It is then natural to introduce the fourth moment

S_{4} (N, W) := \sum_{n, m \in [N, 2 N]} | R_{W} (n / m) |^{4}

since from Hölder’s inequality one now has

V^{2} E_{1} (W) ≪ T^{o (1)} S (N, W)^{1 / 2} S_{4} (N, W)^{1 / 2} + T^{- 50}

(cf. [ 76 , Lemma 3 ] ). The quantity $S_{4} (N, W)$ can also be expressed as

S_{4} (N, W) = \sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W} {| \sum_{n \in [N, 2 N]} n^{- i (t_{1} + t_{2} - t_{3} - t_{4})} |}^{2} .

One can bound this quantity by an $S (N, W)$ type expression:

Lemma 10.5

If $W \subset [- T, T]$ is $1$ -separated and $1 \leq N ≪ T^{O (1)}$ , then one has

S_{4} (N, W) ≪ T^{o (1)} u^{2} S (N, U) + T^{- 100}

for some $1 \leq u ≪ | W |$ and $1$ -separated subset $U$ of $[- 2 T, 2 T]$ with

u | U | ≪ | W |^{2}

and

u^{2} | U | ≪ E_{1} (W) .

This result appears implicitly in [ 76 , p. 229 ] , and is made more explicit in the proof of [ 61 , Lemma 11.6 ] .

Proof ▶

One can bound

S_{4} (N, W) ≪ T^{o (1)} \sum_{t_{1}, t_{2}, t_{3}, t_{4} \in W} \int_{t = t_{1} + t_{2} - t_{3} - t_{4} + O (T^{o (1)})} {| \sum_{n \in [N, 2 N]} n^{- i t} |}^{2} d t + T^{- 100},

and hence

S_{4} (N, W) ≪ T^{o (1)} \sum_{t_{1}, t_{2} \in [- 2 N, 2 N] \cap Z} f (t_{1}) f (t_{2}) \int_{t = t_{1} - t_{2} + O (T^{o (1)})} {| \sum_{n \in [N, 2 N]} n^{- i t} |}^{2} d t + T^{- 100}

where $f$ is the counting function

f (t) := | {(t_{1}, t_{2}) \in W : | t - t_{1} - t_{2} | \leq 1} | .

Note that $f$ is integer valued and bounded above by $| W |$ . By dyadic decomposition, one can then find $1 \leq u ≪ | W |$ and a subset $U$ of $[- 2 N, 2 N] \cap Z$ such that $f (t) ≍ u$ for $t \in U$ , and

S_{4} (N, W) ≪ T^{o (1)} \sum_{t_{1}, t_{2} \in U} u^{2} \int_{t = t_{1} - t_{2} + O (T^{o (1)})} {| \sum_{n \in [N, 2 N]} n^{- i t} |}^{2} d t + T^{- 100}

which we can rearrange as

S_{4} (N, W) ≪ T^{o (1)} u^{2} \int_{s = O (T^{o (1)})} \sum_{n, m \in [N, 2 M]} (n / m)^{i s} | R_{U} (n / m) |^{2} d s + T^{- 100}

and hence by the triangle inequality

S_{4} (N, W) ≪ T^{o (1)} v^{2} S (N, V) + T^{- 100} .

Also, by double counting one easily verifies the claims 7, 8. The claim follows.

10.2 Large value additive energy region

Because the cardinality $| W |$ and additive energy $E_{1} (W)$ of a set $W$ are correlated with each other, as well as with the double zeta sum $S (N, W)$ , we will not be able to consider them separately, and instead we will need to consider the possible joint exponents for these two quantities. We formalize this via the following set:

Definition 10.6 Large value energy region

The large value energy region $E \subset R^{5}$ is defined to be the set of all fixed tuples $(σ, τ, ρ, ρ^{*}, s)$ with $1 / 2 \leq σ \leq 1$ , $τ, ρ, ρ^{'} \geq 0$ , such that there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $V = N^{σ + o (1)}$ , $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ and $S (N, W) = N^{s + o (1)}$ .

We define the zeta large value energy region $E_{ζ} \subset R^{5}$ similarly, but where now $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ is required to be a zeta large value pattern.

Clearly we have

Lemma 10.7 Trivial containment

We have $E_{ζ} \subset E$ .

These regions are related to $LV (σ, τ)$ and ${LV}_{ζ} (σ, τ)$ as follows:

Lemma 10.8

For any fixed $1 / 2 \leq σ \leq 1, τ \geq 0$ , we have

LV (σ, τ) = sup {ρ : (σ, τ, ρ, ρ^{*}, s) \in E}

and

{LV}_{ζ} (σ, τ) = sup {ρ : (σ, τ, ρ, ρ^{*}, s) \in E_{ζ}}

In particular, we have $ρ \leq LV (σ, τ)$ for all $(σ, τ, ρ, ρ^{*}, s) \in E$ , and $ρ \leq {LV}_{ζ} (σ, τ)$ for all $(σ, τ, ρ, ρ^{*}, s) \in E_{ζ}$ .

Proof ▶

Inspired by this, we can define

Definition 10.9

For any fixed $1 / 2 \leq σ \leq 1, τ \geq 0$ , we define

{LV}^{*} (σ, τ) := sup {ρ^{*} : (σ, τ, ρ, ρ^{*}, s) \in E}

and

{LV}_{ζ}^{*} (σ, τ) := sup {ρ^{*} : (σ, τ, ρ, ρ^{*}, s) \in E_{ζ}} .

Thus these exponents are upper bounds for the additive energy of large values of Dirichlet polynomials which may or may not be of zeta function type.

As usual, we have an equivalent non-asymptotic definition of the large value energy region:

Lemma 10.10 Non-asymptotic form of large value energy region

Let $1 / 2 \leq σ \leq 1$ , $τ \geq 0$ , $ρ, ρ^{*} \geq 0$ , and $s \in R$ be fixed. Then the following are equivalent:

$(σ, τ, ρ, ρ^{*}, s) \in E$ .
For every $ε > 0$ and $C > 0$ , there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N \geq C$ , $N^{τ - δ} \leq T \leq N^{τ + δ}$ , $N^{σ - δ} \leq V \leq N^{σ + δ}$ , $N^{ρ - ε} \leq | W | \leq N^{ρ + ε}$ , $N^{ρ^{*} - ε} \leq E_{1} (W) \leq N^{ρ^{*} + ε}$ , and $N^{s - ε} \leq S (N, W) \leq N^{s + ε}$ .

Similarly with $E$ replaced by $E_{ζ}$ , and with $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ required to be a zeta large value pattern.

This lemma is proven by a routine expansion of the definitions, and is omitted.

Lemma 10.11 Basic properties

(Monotonicity in $σ$ ) If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then $(σ^{'}, τ^{'}, ρ, ρ^{*}, s) \in E$ for all $1 / 2 \leq σ^{'} \leq σ$ and $τ^{'} \geq τ$ .
(Subdivision) If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $0 \leq τ^{'} \leq τ$ , then amongst all tuples $(σ, τ^{'}, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ with $ρ^{'} \leq ρ$ , $(ρ^{'})^{*} \leq ρ^{*}$ , and $s^{'} \leq s$ , there exists a tuple with

$ρ \leq ρ^{'} + τ - τ^{'};$

there exists a tuple with

$ρ^{*} \leq ρ^{'} + 3 min (ρ - ρ^{'}, τ - τ^{'});$

and there exists a tuple with

$s \leq s^{'} + 2 min (ρ - ρ^{'}, τ - τ^{'}) .$

(But it may not be the same tuple that satisfies all three properties.)
(Trivial bounds) If $(σ, τ, ρ, ρ^{*}, s) \in E$ , one has

$2 ρ, 4 ρ - τ \leq ρ^{*} \leq 3 ρ .$

Proof ▶

The claim (i) is trivial, so we turn to (ii). By definition, there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ , and $S (N, W) = N^{s + o (1)}$ . We now partition $J$ into $N^{τ - τ^{'} + o (1)}$ subintervals $I$ of length $N^{τ^{'} + o (1)}$ , and subdivide $W$ into $W_{I}$ accordingly. By dyadically pigeonholing, we can then subdivide this collection $I$ of intervals into $N^{o (1)}$ subcollections, where on each subcollection there exists fixed $ρ^{'}, (ρ^{'})^{*}, s^{'}$ such that $| W_{I} | = N^{ρ^{'} + o (1)}$ , $E_{1} (W_{I}) = N^{(ρ^{'})^{*} + o (1)}$ , and $S (N, W_{I}) = N^{s^{'} + o (1)}$ . Since $W_{I} \subset W$ , this forces $ρ^{'} \leq ρ$ , $(ρ^{'})^{*} \leq ρ^{*}$ , and $s^{'} \leq s$ . From Definition 10.6 we see that $(σ, τ^{'}, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ .

By the pigeonhole principle, one of these subcollections must contribute at least $N^{- o (1)}$ of the cardinality of $W$ . Since there are at most $N^{τ - τ^{'} + o (1)}$ intervals in this collection, we must have $ρ \leq ρ^{'} + τ - τ^{'}$ in this case.

By Lemma 10.2(iii), we also know that a (possibly different subcollection) must contribute at least $N^{- o (1)}$ of the additive energy of $W$ . The number of intervals in this subcollection is at most $min (N^{τ - τ^{'} + o (1)}, N^{ρ - ρ^{'} + o (1)})$ . Applying Lemma 10.2(iii) again, we conclude $ρ^{*} \leq (ρ^{*})^{'} + 3 min (ρ - ρ^{'}, τ - τ^{'})$ .

Finally, from 3, we know that a (possibly yet another subcollection) must contribute at least $N^{- o (1)}$ of the double zeta sum $S (N, W)$ . The number of intervals in this subcollection is at most $min (N^{τ - τ^{'} + o (1)}, N^{ρ - ρ^{'} + o (1)})$ . Applying Lemma 3(iii) again, we conclude $s \leq s^{'} + 2 min (ρ - ρ^{'}, τ - τ^{'})$ .

Lemma 10.12 Raising to a power

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , and $k \geq 1$ , then amongst all tuples $(σ, τ / k, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ with $ρ^{'} \leq ρ / k$ , $(ρ^{'})^{*} \leq ρ^{*} / k$ , and $s^{'} \leq s / k$ , there exists a tuple with $ρ^{'} = ρ / k$ , there exists a tuple with $(ρ^{'})^{*} = ρ^{*} / k$ , and there exists a tuple with $s^{'} = s / k$ . (These may be three different tuples.)

Proof ▶

By definition, there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ , and $S (N, W) = N^{s + o (1)}$ . Observe that

{(\sum_{n \in [N, 2 N]} a_{n} n^{- i t})}^{k} = \sum_{n \in [N^{k}, 2^{k} N^{k}]} b_{n} n^{- i t}

for some coefficients $b_{n} = O (n^{o (1)})$ . In particular, partitioning $[N^{k}, 2^{k} N^{k}]$ into $O (1)$ sub-intervals $[N^{'}, 2 N^{'}]$ with $N^{'} = N^{k + o (1)}$ , we can partition $W$ into $O (1)$ subcollections $W_{N^{'}}$ , such that

| \sum_{n \in [N^{'}, 2 N^{'}]} b_{n} n^{- i t} | ≫ V^{k} = (N^{'})^{σ + o (1)}

for all $t \in W_{N^{'}}$ . Again by the pigeonhole principle, one of the $W_{N^{'}}$ must have cardinality $N^{ρ + o (1)}$ , one must have energy $N^{ρ^{*} + o (1)}$ , and one must have double zeta sum $N^{s + o (1)}$ (but these may be different $W_{N^{'}}$ ). Each of these $W_{N^{'}}$ then give the different conclusions to the lemma.

Morally speaking, one should be able to obtain equality in all three conclusions of Lemma 10.12 simultaneously, i.e. that $(σ, τ, ρ, ρ^{*}, s) \in E$ essentially implies $(σ, τ / k, ρ / k, ρ^{*} / k, s / k) \in E$ . This is because in practice one frequently controls $E$ by computing a containment region $E_{1}$ that possesses precisely the required monotonicity property. Specifically, we have

Lemma 10.13 Monotonicity criterion

Let $E_{1}$ be the intersection of sets $E_{i}$ , each of the form

\begin{array}{r} {(σ, τ, ρ, ρ^{*}, s) \in R^{5} : ρ \leq f_{1} (ρ^{*}, s), ρ^{*} \leq f_{2} (ρ, s), s \leq f_{3} (ρ, ρ^{*})} \end{array}

for some monotonically increasing functions $f_{1}, f_{2}, f_{3}$ (that possibly also depend on $σ$ and $τ$ ).

Suppose amongst all tuples $(σ, τ, ρ^{'}, (ρ^{*})^{'}, s^{'}) \in E_{1}$ with $ρ^{'} \leq ρ$ , $(ρ^{*})^{'} \leq ρ^{*}$ and $s^{'} \leq s$ , there exists a tuple with $ρ^{'} = ρ$ , a tuple with $(ρ^{*})^{'} = ρ^{*}$ and a tuple with $s^{'} = s$ (not necessarily the same tuple each time). Then, $(σ, τ, ρ, ρ^{*}, s) \in E_{1}$ .

Proof ▶

Suppose that $(σ, τ, ρ, (ρ^{*})^{'}, s^{'}) \in E_{1}$ for some $(ρ^{*})^{'} \leq ρ^{*}$ and $s \leq s^{'}$ so that also $(σ, τ, ρ, (ρ^{*})^{'}, s^{'}) \in E_{i}$ . Then by definition $ρ \leq f_{1} ((ρ^{*})^{'}, s^{'})$ . Since $f_{1}$ is monotonically increasing (with respect to both $ρ^{*}$ and $s$ ), one has $ρ \leq f_{1} (ρ^{*}, s)$ . Similarly, $(σ, τ, ρ^{'}, ρ^{*}, s^{'}) \in E_{1}$ implies $ρ^{*} \leq f_{2} (ρ, s)$ and $(σ, τ, ρ^{'}, (ρ^{*})^{'}, s) \in E_{1}$ implies $s \leq f_{3} (ρ, ρ^{*})$ , which together imply $(σ, τ, ρ, ρ^{*}, s) \in E_{i}$ by definition. Hence $(σ, τ, ρ, ρ^{*}, s) \in E_{1}$ since $E_{1}$ is the intersection of sets $E_{i}$ .

Lemma 10.14 Raising to a power, alternative formulation

Let $k$ be a positive integer, $E_{1} \subseteq R^{5}$ be a set satisfying the monotonicity criterion of Lemma 10.13 and

E_{k} := {(σ, τ, ρ, ρ^{*}, s) \in R^{5} : (σ, τ / k, ρ / k, ρ^{*} / k, s / k) \in E_{1}} .

If $E \subseteq E_{1}$ then $E \subseteq E_{k}$ .

Proof ▶

Suppose that $(σ, τ, ρ, ρ^{*}, s) \in E \subseteq E_{1}$ . By Lemma 10.12 and Lemma 10.13, $(σ, τ / k, ρ / k, ρ^{*} / k, s / k) \in E_{1}$ , hence by definition $(σ, τ, ρ, ρ^{*}, s) \in E_{k}$ .

10.3 Known relations for the large value energy region

Theorem 10.15 Reflection principle

[ 110 , § 11.5 ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ with $σ \geq 3 / 4$ and $τ > 1$ , then for any integer $k \geq 1$ , either $ρ \leq 2 - 2 σ$ , or there exists $0 < α \leq k (τ - 1)$ and $(σ, τ / α, ρ / α, ρ^{*} / α, s^{'} / α) \in E$ such that

ρ \leq min (2 - 2 σ, k (3 - 4 σ) / 2 + s^{'} - 1) .

Proof ▶

By definition, there exists a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $T = N^{τ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ and $S (N, W) = N^{s + o (1)}$ . By [ 110 , (11.58) ] , one has

| W |^{2} ≪ T^{o (1)} (| W | N^{2 - 2 σ} + N^{1 - 2 σ} | W |^{2} + N^{(3 - 4 σ) / 2} \int_{v = O (T^{o (1)})} \sum_{t, t^{'} \in W} | \sum_{n \leq 4 T / N} n^{- 1 / 2 + i t - i t^{'} + i v} | d v) .

Since $σ > 1 / 2$ , the $N^{1 - 2 σ} | W |^{2}$ term can be dropped. Applying Hölder’s inequality and dyadic pigeonholing as in [ 110 , (11.59) ] , we conclude that

| W | ≪ T^{o (1)} (N^{2 - 2 σ} + N^{k (3 - 4 σ) / 2} {(\sum_{t, t^{'} \in W} {| \sum_{n \in [N^{'}, 2 N^{'}]} b_{n} n^{- 1 / 2 + i t - i t^{'} + i v} |}^{2})}^{1 / 2})

for some $v = O (T^{o (1)})$ and coefficients $b_{n} = O (T^{o (1)})$ , and some $N^{'} ≪ (4 T / N)^{k}$ . After passing to a subsequence if necessary, we may assume that $N^{'} = N^{α + o (1)}$ for some $0 \leq α \leq k (τ - 1)$ . If $α = 0$ then the second term here is negligible compared to the first and we obtain $ρ \leq 2 - 2 σ$ , so suppose that $α > 0$ . Using [ 110 , Lemma 11.1 ] to eliminate the $b_{n} n^{- 1 / 2 + i v}$ coefficients, we conclude that

| W | ≪ T^{o (1)} (N^{2 - 2 σ} + N^{k (3 - 4 σ) / 2 - 1} S (N^{'}, W)) .

By construction, we have $S (N^{'}, W) = (N^{'})^{s^{'} / α + o (1)} = N^{s^{'} + o (1)}$ for some tuple $(σ, τ / α, ρ / α, ρ^{*} / α, s^{'} / α) \in E$ . The claim follows.

Heuristically one expects $s \leq max (ρ + 1, 2 ρ) + 1$ (see [ 110 , (11.63) ] ). There is one easy case in which this is true:

Lemma 10.16

If $(σ, τ, ρ, ρ^{*}, s) \in E$ with $τ < 1$ , then $s \leq max (ρ + 1, 2 ρ) + 1$ .

Recorded in literature.py as:
add_lver_ivic_1985()

Proof ▶

N^{- 1} S (N, W) ≪ T^{ε} (| W | N + | W |^{2})

for any fixed $ε > 0$ , which gives the claim. .

Another bound is

Lemma 10.17

[ 110 , Lemma 11.2 ] If $(k, ℓ)$ is an exponent pair with $k > 0$ , and $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

s \leq max (ρ + 1, 5 ρ / 3 + τ / 3, \frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ} ρ + \frac{k + ℓ}{1 + 2 k + 2 ℓ} τ) + 1.

Proof ▶

N^{- 1} S (N, W) ≪ | W | N + | W |^{5 / 3} T^{1 / 3 + ε} + | W |^{\frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ}} T^{\frac{k + ℓ + ε}{1 + 2 k + 2 ℓ}}

for any fixed $ε > 0$ , which gives the claim.

Finally, we have the useful

Lemma 10.18 Heath-Brown bound on double sums

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

s \leq max (ρ + 1, 2 ρ, 5 ρ / 4 + τ / 2) + 1.

Note that if $τ \leq 3 / 2$ , the $5 ρ / 4 + τ / 2$ term is bounded by the convex combination $(3 / 4) (ρ + 1) + (1 / 4) (2 ρ)$ and may therefore be omitted.

Recorded in literature.py as:
add_lver_heath_brown_1979()

Proof ▶

N^{- 1} S (N, W) ≪ T^{ε} (| W | N + | W |^{2} + | W |^{5 / 4} T^{1 / 2}),

giving the claim.

Lemma 10.5 can be formulated in terms of the large value energy region as follows.

Lemma 10.19

If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then there exists $(σ, τ, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ and $0 \leq κ \leq ρ$ such that

κ + ρ^{'} \leq 2 ρ

2 κ + ρ^{'} \leq ρ^{*}

and

ρ^{*} + 2 σ \leq κ + (s + s^{'}) / 2.

Proof ▶

By definition, there exists a large value pattern

(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)

with $N \geq 1$ unbounded, $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ , and $S (N, W) = N^{s + o (1)}$ . From 6 we have

V^{2} E_{1} (W) ≪ T^{o (1)} S (N, W)^{1 / 2} S_{4} (N, W)^{1 / 2} + T^{- 50} .

By Lemma 10.5, there exists $1 \leq u ≪ | W |$ and a $1$ -separated subset $U$ of $[- 2 T, 2 T]$ such that such that

V^{2} E_{1} (W) ≪ T^{o (1)} u S (N, W)^{1 / 2} S (N, U)^{1 / 2} + T^{- 50}

with 7, 8 holding. Since $W$ is non-empty, $E_{1} (W) \geq 1$ and $V \geq N^{1 / 2} \geq 1$ , so the $T^{- 50}$ error here may be discarded. Passing to a subsequence, we may assume that $u = N^{κ + o (1)}$ for some $0 \leq κ \leq ρ$ , and that $| U | = N^{ρ^{'} + o (1)}$ for some $ρ^{'} \geq 0$ . Then we have $S_{2} (N, U) = s^{'}$ for some $(σ, τ, ρ^{'}, (ρ^{'})^{*}, s^{'}) \in E$ , and the claim follows.

These bounds on the double zeta sums can be used to control additive energies:

Theorem 10.20 Heath-Brown relation

[ 79 , (33) ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ , then one has

ρ^{*} \leq 1 - 2 σ + \frac{1}{2} max (ρ + 1, 2 ρ, \frac{5}{4} ρ + \frac{τ}{2}) + \frac{1}{2} max (ρ^{*} + 1, 4 ρ, \frac{3}{4} ρ^{*} + ρ + \frac{τ}{2}) .

Recorded in literature.py as:
add_lver_heath_brown_1979b1()

Proof ▶

By Lemma 10.19 we have

ρ^{*} + 2 σ \leq κ + (max (ρ + 1, 2 ρ, 5 ρ / 4 + τ / 2) + max (ρ^{'} + 1, 2 ρ^{'}, 5 ρ^{'} / 4 + τ / 2)) / 2 + 1

for some $0 \leq κ \leq ρ$ with

κ + ρ^{'} \leq 2 ρ

2 κ + ρ^{'} \leq ρ^{*}

In particular,

2 κ + 5 ρ^{'} / 4 \leq 3 ρ^{*} / 4 + ρ

and the claim follows after moving the $κ$ inside the second maximum and performing some algebra.

Corollary 10.21 Simplified Heath-Brown relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $τ \leq 3 / 2$ , then

ρ^{*} \leq max (3 ρ + 1 - 2 σ, ρ + 4 - 4 σ, 5 ρ / 2 + (3 - 4 σ) / 2) .

Recorded in literature.py as:
add_lver_heath_brown_1979b2()

This result essentially appears as [ 79 , Lemma 3 ] .

Proof ▶

Apply the previous result. For $τ \leq 3 / 2$ we observe that $5 ρ / 4 + τ / 2$ is less than $5 ρ / 4 + 3 / 4$ , which is a convex combination of $ρ + 1$ and $2 ρ$ . Similarly $3 ρ^{*} / 4 + ρ + τ / 2$ is less than $3 ρ^{*} / 4 + ρ + 3 / 4$ , which is a convex combination of $ρ^{*} + 1$ and $4 ρ$ . We conclude that

ρ^{*} \leq 1 - 2 σ + max (ρ + 1, 2 ρ) / 2 + max (ρ^{*} + 1, 4 ρ) / 2.

Thus $ρ^{*}$ is less than one of

1 - 2 σ + (ρ + ρ^{*} + 2) / 2, 1 - 2 σ + (5 ρ + 1) / 2, 1 - 2 σ + (2 ρ + ρ^{*} + 1) / 2, 1 - 2 σ + (6 ρ) / 2;

solving for $ρ^{*}$ , we conclude

ρ^{*} \leq max (4 - 4 σ + ρ, (3 - 4 σ) / 2 + 5 ρ / 2, 3 - 4 σ + 2 ρ, 1 - 2 σ + 3 ρ) .

But since $σ \geq 1 / 2$ , $3 - 4 σ + 2 ρ$ is less than $5 / 2 - 3 σ + 2 ρ$ , which is the mean of $4 - 4 σ + ρ$ and $1 - 2 σ + 3 ρ$ . Thus

ρ^{*} \leq max (4 - 4 σ + ρ, (3 - 4 σ) / 2 + 5 ρ / 2, 1 - 2 σ + 3 ρ),

which gives the claim.

Similarly, using Lemma 10.17 and Lemma 10.19, one has

Theorem 10.22

If $(k, ℓ)$ be an exponent pair with $k > 0$ and $(σ, τ, ρ, ρ^{*}, s) \in E$ , then

\begin{aligned} ρ^{*} \leq 1 - 2 σ & + \frac{1}{2} max (ρ + 1, \frac{5}{3} ρ + \frac{τ}{3}, \frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ} ρ + \frac{k + ℓ}{1 + 2 k + 2 ℓ} τ) \\ + \frac{1}{2} max (ρ^{*} + 1, 4 ρ, \frac{3}{4} ρ^{*} + ρ + \frac{τ}{2}) . \end{aligned}

Implemented at additive_energy.py as:
ep_to_lver(eph)

Proof ▶

By Lemma 10.19 and Lemma 10.17, there exists some $(σ, τ, ρ^{'}, (ρ^{*})^{'}, s^{'}) \in E$ satisfying

0 \leq κ \leq ρ, κ + ρ^{'} \leq 2 ρ, 2 κ + ρ^{'} \leq ρ^{*}

and

\begin{aligned} ρ^{*} + 2 σ \leq κ & + \frac{1}{2} max (ρ + 1, 5 ρ / 3 + τ / 3, \frac{2 + 3 k + 4 ℓ}{1 + 2 k + 2 ℓ} ρ + \frac{k + ℓ}{1 + 2 k + 2 ℓ} τ) \\ + \frac{1}{2} max (ρ^{'} + 1, 2 ρ^{'}, \frac{5}{4} ρ^{'} + \frac{τ}{2}) + 1. \end{aligned}

The result follows by moving the $κ$ term into the second maximum and applying 9.

Lemma 10.23 Second Heath-Brown relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ \leq max (2 - 2 σ, ρ^{*} / 4 + max (τ / 4 + k (3 - 4 σ) / 4, k τ / 4 + k (1 - 2 σ) / 2))

for any positive integer $k$ .

Recorded in literature.py as:
add_lver_heath_brown_1979c(K)

Proof ▶

| W | ≪ T^{ε} (N^{2 - 2 σ} + E_{1} (W)^{1 / 4} (T^{1 / 4} N^{k (3 - 4 σ) / 4} + T^{k / 4} N^{k (1 - 2 σ) / 2}))

for any fixed $ε > 0$ , giving the claim.

Lemma 10.24 Guth-Maynard relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ \leq max (2 - 2 σ, 1 - 2 σ + max (S_{1}, S_{2}, S_{3}) / 3)

where $S_{1}, S_{2}, S_{3}$ are real numbers with

S_{1} \leq - 10,

S_{2} \leq max (2 + 2 ρ, τ + 1 + (2 - 1 / k) ρ, 2 + 2 ρ + (τ / 2 - 3 ρ / 4) / k)

for any positive integer $k$ and

S_{3} \leq 2 τ + ρ / 2 + ρ^{*} / 2

and also

S_{3} \leq max (2 τ + 3 ρ / 2, τ + 1 + ρ / 2 + ρ^{*} / 2) .

Recorded in literature.py as:
add_lver_guth_maynard_2024a()

Proof ▶

| W | ≪ N^{2 - 2 σ} + N^{1 - 2 σ} (S_{1} + S_{2} + S_{3})^{1 / 3}

for three expressions $S_{1}, S_{2}, S_{3}$ defined after [ 61 , (5.5) ] . From [ 61 , Proposition 5.1 ] we have

S_{1} ≪ T^{- 10} .

From [ 61 , Proposition 6.1 ] we have

S_{2} ≪ T^{o (1)} (N^{2} | W |^{2} + T N | W |^{2 - 1 / k} + N^{2} | W |^{2} (\frac{T^{1 / 2}}{| W |^{3 / 4}})^{1 / k}) .

From [ 61 , Proposition 8.1 ] we have

S_{3} ≪ T^{2 + o (1)} | W |^{1 / 2} E_{1} (W)^{1 / 2}

while from [ 61 , Proposition 10.1 ] we have

S_{3} ≪ T^{2 + o (1)} | W |^{3 / 2} + T^{1 + o (1)} N | W |^{1 / 2} E_{1} (W)^{1 / 2} .

Combining all these bounds, we obtain the claim.

Lemma 10.25 Second Guth-Maynard relation

[ 61 , Lemma 1.7 ] If $(σ, τ, ρ, ρ^{*}, s) \in E$ then

ρ^{*} \leq ρ + s - 2 σ .

In particular, from Lemma 10.18 we see for $τ \leq 3 / 2$ that

ρ^{*} \leq max (3 ρ + 1 - 2 σ, 2 ρ + 2 - 2 σ) .

Recorded in literature.py as:
add_lver_guth_maynard_2024b()

Proof ▶

By definition, we can find a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, W)$ with $N > 1$ unbounded, $T = N^{τ + o (1)}$ , $V = N^{σ + o (1)}$ , $| W | = N^{ρ + o (1)}$ , $E_{1} (W) = N^{ρ^{*} + o (1)}$ , and $S (N, W) = N^{s + o (1)}$ . From 5 one has

V^{2} E_{1} (W) ≪ T^{o (1)} | W | S (N, W) + T^{- 50} .

Since $W$ is non-empty, $E_{1} (W) \geq 1$ , and $V ≫ 1$ , so the $T^{- 50}$ error can be discarded. The claim then follows.

Lemma 10.26 Third Guth-Maynard relation

If $(σ, τ, ρ, ρ^{*}, s) \in E$ and $1 \leq τ \leq 4 / 3$ , then

ρ^{*} \leq max (ρ + 4 - 4 σ, 21 ρ / 8 + τ / 4 + 1 - 2 σ, 3 ρ + 1 - 2 σ) .

Recorded in literature.py as:
add_lver_guth_maynard_2024c()

Proof ▶

E_{1} (W) ≪ T^{o (1)} (| W | N^{4 - 4 σ} + | W |^{21 / 8} T^{1 / 4} N^{1 - 2 σ} + | W |^{3} N^{1 - 2 σ}),

giving the claim.

We can put this all together to prove the Guth–Maynard large values theorem.

Theorem 10.27 Guth–Maynard large values theorem

[ 61 , Theorem 1.1 ] One has

LV (σ, τ) \leq max (2 - 2 σ, 18 / 5 - 4 σ, τ + 12 / 5 - 4 σ) .

Recorded in literature.py as:
add_guth_maynard_large_values_estimate()
Derived in derived.py as:
prove_guth_maynard_large_values_theorem()
prove_guth_maynard_lvt_from_intermediate_lemmas()

Proof ▶

For $σ \leq 7 / 10$ this follows from Lemma 7.9, and for $σ \geq 8 / 10$ it follows from Lemma 7.12. Thus we may assume that $7 / 10 \leq σ \leq 8 / 10$ . By subdivision (Lemma 7.7) it then suffices to treat the case $τ = 6 / 5$ , that is to say to show that

ρ \leq 18 / 5 - 4 σ

whenever $(σ, τ, ρ, ρ^{*}, s) \in E$ with $τ = 6 / 5$ and $7 / 10 \leq σ \leq 8 / 10$ .

Applying Lemma 10.24 and discarding the very negative $S_{1}$ term, we have

ρ \leq max (2 - 2 σ, 1 - 2 σ + max (S_{2}, S_{3}) / 3)

where $S_{2}, S_{3}$ are real numbers with

S_{2} \leq max (2 + 2 ρ, τ + 1 + (2 - 1 / k) ρ, 2 + 2 ρ + (τ / 2 - 3 ρ / 4) / k)

for any positive integer $k$ and

S_{3} \leq 2 τ + ρ / 2 + ρ^{*} / 2

and also

S_{3} \leq max (2 τ + 3 ρ / 2, τ + 1 + ρ / 2 + ρ^{*} / 2) .

From the latter bound and Lemma 10.26, one has

S_{3} \leq max (2 τ + 3 ρ / 2, τ + ρ + 3 - 2 σ, τ + 2 ρ + 3 / 2 - σ, 9 τ / 8 + 29 ρ / 16 + 3 / 2 - σ) .

Inserting this and the $S_{2}$ bound (with $k = 4$ ) into the bound for $ρ$ and simplifying (using $τ = 6 / 5$ ), we eventually obtain the desired bound $ρ \leq 18 / 5 - 4 σ$ .

We also record a variant of that theorem from the same paper:

Theorem 10.28 Additional Guth–Maynard large values estimate

For any $1 / 2 < σ < 1$ , $τ \geq 1$ , and natural number $k$ one has

\begin{aligned} LV (σ, τ) \leq max ( & 2 - 2 σ, 5 - 6 σ, (4 - 6 σ) \frac{k}{k + 1} + \frac{k}{k + 1} τ, (5 - 6 σ) \frac{4 k}{4 k + 3} + \frac{2}{4 k + 3} τ, \\ 2 - 4 σ + \frac{4}{3} τ, 3 - 4 σ + \frac{τ}{2}, \frac{9}{2} - 7 σ + τ, \frac{72 - 112 σ}{19} + \frac{18}{19} τ) . \end{aligned}

If one specializes to the case $σ \geq 7 / 10$ and $1 \leq τ \leq 6 / 5$ , one then has

LV (σ, τ) \leq max (2 - 2 σ, 3 - 4 σ + \frac{τ}{2}, (4 - 6 σ) \frac{k}{k + 1} + \frac{k}{k + 1} τ, (5 - 6 σ) \frac{4 k}{4 k + 3} + \frac{2}{4 k + 3} τ) .

and also

LV (σ, τ) \leq max (2 - 2 σ, 3 - 4 σ + \frac{τ}{2}, \frac{46 - 60 σ}{5} + \frac{30 σ - 21}{5} τ) .

Recorded in literature.py as:
add_guth_maynard_large_values_estimate2(Constants.LARGE_VALUES_TRUNCATION)
Derived in derived.py as:
prove_guth_maynard_intermediate_lvt()
prove_guth_maynard_intermediate_lvt2()

Proof ▶

(Sketch) The first bound 10 is [ 61 , (12.1) ] , and is proven by the same methods used to prove Theorem 10.27. The second and third bounds 13, 14 are in [ 61 , Proposition 12.1 ] . The second bound 13 follows from the $k = 3$ case of Theorem 7.16 when $σ \geq 39 / 50$ , and from 10 together with Theorems 7.9, 7.12 for $σ < 39 / 50$ . The third bound 14 follows from Theorems 7.9, 7.12 when $τ \leq 4 - 4 σ$ , and for $τ > 4 - 4 σ$ the bound follows from 13 after optimizing in $k$ (see the proof of [ 61 , Proposition 12.1 ] for details).

Now we turn to another application of double zeta sums to large value theorems.

Theorem 10.29 Bourgain large values theorem

[ 20 ] Let $1 / 2 < σ < 1$ and $τ > 0$ , and let $ρ := LV (σ, τ)$ . Let $α_{1}, α_{2} \geq 0$ be real numbers. Then either

ρ \leq max (α_{2} + 2 - 2 σ, - α_{2} + 2 τ + 4 - 8 σ, - 2 α_{1} + τ + 12 - 16 σ)

or else there exists $s \geq 0$ such that

\begin{aligned} \frac{1}{2} max (ρ + 2, 2 ρ + 1, 5 ρ / 4 + τ / 2 + 1) + \frac{1}{2} max (s + 2, 2 s + 1, 5 s / 4 + τ / 2 + 1) \geq \\ max (- 2 α_{1} + 2 σ + s + ρ, - α_{1} - α_{2} / 2 + 2 σ + s / 2 + 3 ρ / 2) . \end{aligned}

Proof ▶

By Definition 7.2, we can find a large value pattern $(N, T, V, (a_{n})_{n \in [N, 2 N]}, J, R)$ with $N > 1$ unbounded, $N \geq 1$ , $T = N^{τ + o (1)}$ , $| R | = N^{ρ + o (1)}$ , $V = N^{σ + o (1)}$ ; we use $R$ here instead of $W$ to be consistent with the notation from [ 20 ] . Now set $δ_{1} := N^{- α_{1}}$ , $δ_{2} := N^{- α_{2}}$ . From [ 20 , (4.41), (4.42) ] , one has the inequality

| R | \leq | R^{(1)} | + | R^{(2)} |

for certain sets $R^{(1)}$ and $R^{(2)}$ with the former set obeying the bound

| R^{(1)} | ≪ δ_{2}^{- 1} N^{2} V^{- 2} + δ_{2} T^{2} N^{4} V^{- 8} + δ_{1}^{2} T N^{12} V^{- 16} .

Hence, we either have

| R | ≪ δ_{2}^{- 1} N^{2} V^{- 2} + δ_{2} T^{2} N^{4} V^{- 8} + δ_{1}^{2} T N^{12} V^{- 16}

which implies 15, or else

| R | ≪ | R^{(2)} | .

Henceforth we assume that 19 holds. From [ 20 , (4.53), (4.54) ] we may upper bound

T^{- ε} δ^{'} (δ^{″})^{2} V^{2} | S | | R^{(2)} | + T^{- ε} δ_{1} V^{2} | S |^{1 / 2} \sum_{α} | R_{α} |^{3 / 2}

≪ T^{ε} S (N, R^{(2)})^{1 / 2} S (N, S)^{1 / 2}

for arbitrarily small fixed $ε$ , some $δ^{'}, δ^{″} > 0$ with $δ^{'} > T^{- ε} (δ_{1} / δ^{″})^{2}$ (see [ 20 , (4.37) ] ), some set $S$ (which will be non-empty by [ 20 , (4.47) ] ), and some sets $R_{α}$ defined in [ 20 , (4.39) ] , where the double zeta sums $S (N, W)$ are defined in 2. Applying Lemma 10.18, the latter expression is bounded by

≪ T^{ε} (| R | N^{2} + | R |^{2} N + | R |^{5 / 4} T^{1 / 2} N)^{1 / 2} (| S | N^{2} + | S |^{2} N + | S |^{5 / 4} T^{1 / 2} N)^{1 / 2} .

Meanwhile, from [ 20 , (4.57) ] , the expression 20 is bounded from below by

≫ T^{- 2 ε} (δ_{1}^{2} V^{2} | S | | R | + δ_{1} δ_{2}^{1 / 2} V^{2} | S |^{1 / 2} | R |^{3 / 2}) .

After passing to a subsequence, we can ensure that $| S | = N^{s + o (1)}$ for some $s > 0$ . Combining these bounds and writing all expressions as powers of $N$ , we obtain the claim (after sending $ε \to 0$ ).

Corollary 10.30 Bourgain large values theorem, simplified version

[ 20 , Lemma 4.60 ] Let the notation be as above, but additionally assume $ρ \leq min (1, 4 - 2 τ)$ . Then

ρ \leq max (α_{2} + 2 - 2 σ, α_{1} + α_{2} / 2 + 2 - 2 σ, - α_{2} + 2 τ + 4 - 8 σ, - 2 α_{1} + τ + 12 - 16 σ, 4 α_{1} + 2 + max (1, 2 τ - 2) - 4 σ) .

In [ 20 ] this bound is only established in the case $τ \leq 3 / 2$ (in which case the condition on $ρ$ simplifies to $ρ \leq 1$ , and the final term $4 α_{1} + 2 + max (1, 2 τ - 2) - 4 σ$ simplifies to $4 α_{1} + 3 - 4 σ$ ), but the argument extends to the $τ > 3 / 2$ case without significant difficulty.

Proof ▶

With $ρ \leq min (1, 4 - 2 τ)$ , $5 ρ / 4 + τ / 2 + 1$ and $2 ρ + 1$ are both bounded by $ρ + 2$ , hence

max (ρ + 2, 2 ρ + 1, 5 ρ / 4 + τ / 2 + 1) = ρ + 2.

Furthermore, $5 s / 4 + τ + 1$ is a convex combination of $s + 2$ and $2 s + 2 τ - 2$ , hence

max (s + 2, 2 s + 1, 5 s / 4 + τ / 2 + 1) \leq max (s + 2, 2 s + max (1, 2 τ - 2)) .

Thus 16 simplifies to

(ρ + 2) / 2 + max (s + 2, 2 s + max (1, 2 τ - 2)) / 2 \geq max (- 2 α_{1} + 2 σ + s + ρ, - α_{1} - α_{2} / 2 + 2 σ + s / 2 + 3 ρ / 2) .

Thus either

(ρ + 2) / 2 + (s + 2) / 2 \geq - α_{1} - α_{2} / 2 + 2 σ + s / 2 + 3 ρ / 2

(ρ + 2) / 2 + (2 s + max (1, 2 τ - 2)) / 2 \geq - 2 α_{1} + 2 σ + s + ρ .

In both cases we may eliminate $s$ and solve for $ρ$ to obtain

ρ \leq α_{1} + α_{2} / 2 + 2 - 2 σ

ρ \leq 4 α_{1} + 2 + max (1, 2 τ - 2) - 4 σ,

giving the claim.

With the aid of computer assistance, one is able to produce an optimized version of the above large values theorem. We have

Corollary 10.31 Bourgain large values theorem, optimized version

For each row $(ρ_{0}, α_{1}, α_{2}, S)$ of Table 10.1, one has

ρ \leq ρ_{0} (σ, τ), (σ, τ) \in S .

Recorded in literature.py as:
add_bourgain_large_values_estimate()

Derived in derived.py as:
prove_bourgain_large_values_theorem()

Proof ▶

Follows from substituting the specified values of $α_{1}$ and $α_{2}$ and a routine calculation. □

Table 10.1 Bounds on

LV (σ, τ)

for

1 / 2 \leq σ \leq 1

and

τ \geq 1

$ρ_{0} (σ, τ)$	$α_{1}$	$α_{2}$	$S$
$\frac{16}{3} - \frac{20}{3} σ + \frac{1}{3} τ$	$\frac{10}{3} - \frac{14}{3} σ + \frac{τ}{3}$	$0$	$\begin{array}{r} - 1 + τ \geq 0, \\ 10 - 14 σ + τ \geq 0, \\ 4 + 4 σ - 5 τ \geq 0, \\ - 11 + 16 σ - τ \geq 0. \end{array}$
$5 - 7 σ + \frac{3}{4} τ$	$\frac{7}{2} - \frac{9}{2} σ + \frac{τ}{8}$	$- 1 - σ + \frac{5}{4} τ$	$\begin{array}{r} 8 - 8 σ - τ \geq 0, \\ - 16 + 20 σ + \frac{1}{3} τ \geq 0, \\ - 6 + 10 σ - \frac{7}{6} τ \geq 0, \\ - 4 - 4 σ + 5 τ \geq 0. \end{array}$
$3 - 5 σ + τ$	$\frac{9}{2} - \frac{11}{2} σ$	$1 - 3 σ + τ$	$\begin{aligned} - 8 + 8 σ + τ & \geq 0, \\ 2 - 6 σ + 2 τ & \geq 0, \\ - 10 + 14 σ - \frac{2}{3} τ & \geq 0, \\ 6 - 2 σ - 2 τ & \geq 0. \end{aligned}$
$- 4 σ + 2 τ$	$0$	$1 - 3 σ + τ$	$\begin{aligned} - 6 + 2 σ + 2 τ & \geq 0, \\ - 12 + 12 σ + τ & \geq 0, \\ 1 - σ & \geq 0. \end{aligned}$
$8 - 12 σ + \frac{4}{3} τ$	$2 - 2 σ - \frac{1}{6} τ$	$6 - 10 σ + \frac{4}{3} τ$	$\begin{array}{r} 15 - 21 σ + τ \geq 0, \\ 12 - 12 σ - τ \geq 0, \\ - \frac{3}{2} + τ \geq 0, \\ 6 - 10 σ + \frac{7}{6} τ \geq 0. \end{array}$
$2 - 2 σ$	$0$	$0$	$\begin{array}{r} 1 - σ \geq 0, \\ - 10 + 14 σ - τ \geq 0, \\ - 1 + τ \geq 0, \\ - 2 + 6 σ - 2 τ \geq 0. \end{array}$
$9 - 12 σ + \frac{2}{3} τ$	$\frac{3}{2} - 2 σ + \frac{1}{6} τ$	$11 - 16 σ + τ$	$\begin{array}{r} \frac{3}{2} - τ \geq 0, \\ - \frac{1}{2} + σ \geq 0, \\ - 1 + τ \geq 0, \\ 11 - 16 σ + τ \geq 0, \\ 16 - 20 σ - \frac{1}{3} τ \geq 0. \end{array}$

The preprint of Kerr [ 130 ] contains additional large value theorems:

Lemma 10.32 Kerr large values theorem

[ 130 , Theorem 2 ] Let $3 / 4 < σ \leq 1$ , $0 \leq τ \leq 3 / 2$ , and $0 \leq ρ \leq LV (σ, τ), 1$ be fixed. Then for any fixed $α \geq 0$ , one has

$ρ \leq max (2 - 2 σ + α, 2 τ + 4 - 8 σ - α, τ / 3 + 16 / 3 - 20 σ / 3 + α / 3, 2 τ / 3 + 9 - 12 σ) .$
[ 130 , Theorem 3 ] Under the same hypotheses as (i), we have for any fixed integer $k \geq 2$ obeying $- α < 4 k σ - (τ + 3 k - 1)$ and $- α < 1 + \frac{1}{k - 1} - τ$ that

$ρ \leq max (2 - 2 σ + α, τ / 3 + (3 k + 4) / 3 - (4 k + 4) σ / 3 + α / 3) .$
[ 130 , Theorem 4 ] If $25 / 32 < σ \leq 1$ , $τ \geq 0$ , and $0 \leq ρ \leq LV (σ, τ), 1, 4 - 2 τ$ are fixed, then for any fixed $α$ with

$26 - 32 σ - τ < - α < 16 σ - 11 - τ$

one has

$ρ \leq max (2 - 2 σ + α, 2 τ + 4 - 8 σ - α, - τ + 8 - 8 σ + 2 α, 10 - 12 σ + 2 α / 3) .$
[ 130 , Theorem 5 ] If $1 / 2 < σ \leq 1$ , $0 \leq τ \leq 3 / 2$ , and $0 \leq ρ \leq LV (σ, τ), 1$ are fixed, then for any fixed $α$ with

$- α < - τ + 8 σ - 5$

one has

$ρ \leq max (2 - 2 σ + α, 4 τ / 3 + 23 / 3 - 12 σ - 2 α / 3, 2 τ / 3 + 14 / 3 - 20 / 3) .$