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

Every quadruple contributing to \(E_1(W)\) then contributes \(\gg 1\) to the right-hand side, giving the upper bound. To get the matching lower bound, note that

and hence

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 \(\varphi \), so that the right-hand side is bounded by

which is then bounded by \(O(E_1(W))\) by choosing the support of \(\varphi \) 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

hence by Cauchy–Schwarz

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

which by Cauchy–Schwarz is bounded by

Thus we see that

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

and the claim follows.

We remark that (v) can also be proven using [ 42 , Lemma 4.8, (4.2) ] .

The left-hand side of 4 can be rewritten as

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

By hypothesis, we have

By standard Fourier arguments (see [ 88 , Lemma 11.3 ] ), we can bound

Since each \(t_1,t_2,t_3\) generates at most \(O(1)\) choices for \(t_4\), we conclude that

The right-hand side can be rewritten as

and the claim then follows from the triangle inequality.

One can bound

and hence

where \(f\) is the counting function

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

which we can rearrange as

and hence by the triangle inequality

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

Clear from definition.

The claim (i) is trivial, so we turn to (ii). By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N {\gt} 1\) unbounded, \(T = N^{\tau +o(1)}\), \(V = N^{\sigma +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\), and \(S(N,W) = N^{s+o(1)}\). We now partition \(J\) into \(N^{\tau -\tau '+o(1)}\) subintervals \(I\) of length \(N^{\tau '+o(1)}\), 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 \(\rho ', (\rho ')^*, s'\) such that \(|W_I| = N^{\rho '+o(1)}\), \(E_1(W_I) = N^{(\rho ')^* + o(1)}\), and \(S(N,W_I) = N^{s'+o(1)}\). Since \(W_I \subset W\), this forces \(\rho ' \leq \rho \), \((\rho ')^* \leq \rho ^*\), and \(s' \leq s\). From Definition 10.6 we see that \((\sigma ,\tau ', \rho ', (\rho ')^*,s') \in \mathcal{E}\).

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 \(N^{\tau -\tau '+o(1)}\) intervals in this collection, we must have \(\rho \leq \rho ' + \tau - \tau '\) 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 \(\min ( N^{\tau -\tau '+o(1)}, N^{\rho -\rho '+o(1)})\). Applying Lemma 10.2(iii) again, we conclude \(\rho ^* \leq (\rho ^*)' + 3\min (\rho -\rho ',\tau -\tau ')\).

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 \(\min ( N^{\tau -\tau '+o(1)}, N^{\rho -\rho '+o(1)})\). Applying Lemma 3(iii) again, we conclude \(s \leq s' + 2\min (\rho -\rho ',\tau -\tau ')\).

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N {\gt} 1\) unbounded, \(T = N^{\tau +o(1)}\), \(V = N^{\sigma +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\), and \(S(N,W) = N^{s+o(1)}\). Observe that

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

for all \(t \in W_{N'}\). Again by the pigeonhole principle, one of the \(W_{N'}\) must have cardinality \(N^{\rho +o(1)}\), one must have energy \(N^{\rho ^*+o(1)}\), 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.

Suppose that \((\sigma , \tau , \rho , (\rho ^*)', s') \in \mathcal{E}_1\) for some \((\rho ^*)'\le \rho ^*\) and \(s \le s'\) so that also \((\sigma , \tau , \rho , (\rho ^*)', s') \in E_i\). Then by definition \(\rho \le f_1((\rho ^*)', s')\). Since \(f_1\) is monotonically increasing (with respect to both \(\rho ^*\) and \(s\)), one has \(\rho \le f_1(\rho ^*, s)\). Similarly, \((\sigma , \tau , \rho ', \rho ^*, s') \in \mathcal{E}_1\) implies \(\rho ^* \le f_2(\rho , s)\) and \((\sigma , \tau , \rho ', (\rho ^*)', s) \in \mathcal{E}_1\) implies \(s \le f_3(\rho , \rho ^*)\), which together imply \((\sigma , \tau , \rho , \rho ^*, s) \in E_i\) by definition. Hence \((\sigma , \tau , \rho , \rho ^*, s) \in \mathcal{E}_1\) since \(\mathcal{E}_1\) is the intersection of sets \(E_i\).

Suppose that \((\sigma , \tau , \rho , \rho ^*, s) \in \mathcal{E}\subseteq \mathcal{E}_1\). By Lemma 10.12 and Lemma 10.13, \((\sigma , \tau /k, \rho /k, \rho ^*/k, s/k) \in \mathcal{E}_1\), hence by definition \((\sigma , \tau , \rho , \rho ^*, s) \in \mathcal{E}_k\).

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). By [ 144 , (11.58) ] , one has

Since \(\sigma {\gt}1/2\), the \(N^{1-2\sigma } |W|^2\) term can be dropped. Applying Hölder’s inequality and dyadic pigeonholing as in [ 144 , (11.59) ] , we conclude that

for some \(v = O(T^{o(1)})\) and coefficients \(b_n = O(T^{o(1)})\), and some \(N' \ll (4T/N)^k\). After passing to a subsequence if necessary, we may assume that \(N' = N^{\alpha +o(1)}\) for some \(0 \leq \alpha \leq k(\tau -1)\). If \(\alpha =0\) then the second term here is negligible compared to the first and we obtain \(\rho \leq 2-2\sigma \), so suppose that \(\alpha {\gt} 0\). Using [ 144 , Lemma 11.1 ] to eliminate the \(b_n n^{-1/2+iv}\) coefficients, we conclude that

By construction, we have \(S(N',W) = (N')^{s'/\alpha +o(1)} = N^{s'+o(1)}\) for some tuple \((\sigma , \tau /\alpha , \rho /\alpha , \rho ^*/\alpha , s'/\alpha ) \in \mathcal{E}\). The claim follows.

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). By the discussion after [ 144 , (11.63) ] , we have

for any fixed \(\varepsilon {\gt}0\), which gives the claim. .

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). From [ 144 , Lemma 11.2 ] we have

for any fixed \(\varepsilon {\gt}0\), which gives the claim.

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). From [ 106 , Theorem 1 ] or [ 144 , Lemma 11.5 ] , one has

giving the claim.

By definition, there exists a large value pattern

with \(N \geq 1\) unbounded, \(T = N^{\tau +o(1)}\), \(V = N^{\sigma +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\), and \(S(N,W) = N^{s+o(1)}\). From 6 we have

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

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 \(u = N^{\kappa +o(1)}\) for some \(0 \leq \kappa \leq \rho \), and that \(|U| = N^{\rho '+o(1)}\) for some \(\rho ' \geq 0\). Then we have \(S_2(N,U) = s'\) for some \((\sigma ,\tau ,\rho ',(\rho ')^*,s') \in \mathcal{E}\), and the claim follows.

By Lemma 10.19 we have

for some \(0 \leq \kappa \leq \rho \) with

In particular,

and the claim follows after moving the \(\kappa \) inside the second maximum and performing some algebra.

Apply the previous result. For \(\tau \leq 3/2\) we observe that \(5\rho /4+\tau /2\) is less than \(5\rho /4 + 3/4\), which is a convex combination of \(\rho +1\) and \(2\rho \). Similarly \(3\rho ^*/4+\rho +\tau /2\) is less than \(3\rho ^*/4+\rho +3/4\), which is a convex combination of \(\rho ^*+1\) and \(4\rho \). We conclude that

Thus \(\rho ^*\) is less than one of

solving for \(\rho ^*\), we conclude

But since \(\sigma \geq 1/2\), \(3-4\sigma + 2\rho \) is less than \(5/2-3\sigma + 2\rho \), which is the mean of \(4-4\sigma +\rho \) and \(1-2\sigma +3\rho \). Thus

which gives the claim.

By Lemma 10.19 and Lemma 10.17, there exists some \((\sigma , \tau , \rho ', (\rho ^*)', s') \in \mathcal{E}\) satisfying

and

The result follows by moving the \(\kappa \) term into the second maximum and applying 9.

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). From [ 107 , Lemma 4 ] , we have

for any fixed \(\varepsilon {\gt}0\), giving the claim.

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). By [ 88 , Proposition 4.6, (5.5) ] , one may bound

for three expressions \(S_1, S_2, S_3\) defined after [ 88 , (5.5) ] . From [ 88 , Proposition 5.1 ] we have

From [ 88 , Proposition 6.1 ] we have

From [ 88 , Proposition 8.1 ] we have

while from [ 88 , Proposition 10.1 ] we have

Combining all these bounds, we obtain the claim.

By definition, we can find a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(V = N^{\sigma +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\), and \(S(N,W) = N^{s+o(1)}\). From 5 one has

Since \(W\) is non-empty, \(E_1(W) \geq 1\), and \(V \gg 1\), so the \(T^{-50}\) error can be discarded. The claim then follows.

By definition, there exists a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N{\gt}1\) unbounded, \(T = N^{\tau +o(1)}\), \(|W| = N^{\rho +o(1)}\), \(E_1(W) = N^{\rho ^*+o(1)}\) and \(S(N,W) = N^{s+o(1)}\). Applying [ 88 , Proposition 11.1 ] , we conclude that

giving the claim.

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

whenever \((\sigma ,\tau ,\rho ,\rho ^*,s) \in \mathcal{E}\) with \(\tau =6/5\) and \(7/10 \leq \sigma \leq 8/10\).

Applying Lemma 10.24 and discarding the very negative \(S_1\) term, we have

where \(S_2, S_3\) are real numbers with

for any positive integer \(k\) and

and also

From the latter bound and Lemma 10.26, one has

Inserting this and the \(S_2\) bound (with \(k=4\)) into the bound for \(\rho \) and simplifying (using \(\tau =6/5\)), we eventually obtain the desired bound \(\rho \leq 18/5-4\sigma \).

(Sketch) The first bound 10 is [ 88 , (12.1) ] , and is proven by the same methods used to prove Theorem 10.27. The second and third bounds 13, 14 are in [ 88 , Proposition 12.1 ] . The second bound 13 follows from the \(k=3\) case of Theorem 7.16 when \(\sigma \geq 39/50\), and from 10 together with Theorems 7.9, 7.12 for \(\sigma {\lt} 39/50\). The third bound 14 follows from Theorems 7.9, 7.12 when \(\tau \leq 4-4\sigma \), and for \(\tau {\gt} 4-4\sigma \) the bound follows from 13 after optimizing in \(k\) (see the proof of [ 88 , Proposition 12.1 ] for details).

By Definition 7.2, we can find a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,R)\) with \(N{\gt}1\) unbounded, \(N \geq 1\), \(T = N^{\tau +o(1)}\), \(|R| = N^{\rho +o(1)}\), \(V = N^{\sigma +o(1)}\); we use \(R\) here instead of \(W\) to be consistent with the notation from [ 21 ] . Now set \(\delta _1 := N^{-\alpha _1}\), \(\delta _2 := N^{-\alpha _2}\). From [ 21 , (4.41), (4.42) ] , one has the inequality

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

Hence, we either have

which implies 15, or else

Henceforth we assume that 19 holds. From [ 21 , (4.53), (4.54) ] we may upper bound

by

for arbitrarily small fixed \(\varepsilon \), some \(\delta ',\delta ''{\gt}0\) with \(\delta ' {\gt} T^{-\varepsilon } (\delta _1/\delta '')^2\) (see [ 21 , (4.37) ] ), some set \(S\) (which will be non-empty by [ 21 , (4.47) ] ), and some sets \(R_\alpha \) defined in [ 21 , (4.39) ] , where the double zeta sums \(S(N,W)\) are defined in 2. Applying Lemma 10.18, the latter expression is bounded by

Meanwhile, from [ 21 , (4.57) ] , the expression 20 is bounded from below by

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

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

Furthermore, \(5s/4+\tau /2+1\) is a convex combination of \(s+2\) and \(2s + 2\tau -2\), hence

Thus 16 simplifies to

Thus either

or

In both cases we may eliminate \(s\) and solve for \(\rho \) to obtain

or

giving the claim.

Follows from substituting the specified values of \(\alpha _1\) and \(\alpha _2\) and a routine calculation.