All claims are clear except perhaps for the upper bound

but this follows because any interval of length \(N^{\tau '+o(1)}\) may be subdivided into \(N^{\tau '-\tau +o(1)}\) intervals of length \(N^{\tau +o(1)}\), so on applying Definition 7.2 to each subinterval and summing (using Lemma 2.1 to ensure uniformity), one obtains the claim.

In view of Lemma 7.4(ii), it suffices to show that \(\mathrm{LV}(\sigma , 2-2\sigma ) \geq 2-2\sigma \). By definition, it suffices to find a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) is a large value pattern with \(N\) unbounded, \(V=N^{\sigma +o(1)}\), \(T = N^{2-2\sigma +o(1)}\), and \(|W| \gg N^{2-2\sigma -o(1)}\).

In the endpoint case \(\sigma = 1\) one can achieve this by setting \(a_n=1\) for all \(n\) and taking \(W=\{ 0\} \), so now we assume that \(1/2 {\lt} \sigma {\lt}1\).

We use the probabilistic method. We divide \([N,2N]\) into \(\asymp N^{2-2\sigma }\) intervals \(I\) of length \(\asymp N^{2\sigma -1}\). On each interval \(I\), we choose \(a_n\) to equal some randomly chosen sign \(\epsilon _I \in \{ -1,+1\} \), with the \(\epsilon _I\) chosen independently in \(I\). If \(t = o(N^{2-2\sigma })\), then \(\sum _{n \in I} a_n n^{-it}\) is equal to \(\epsilon _I\) times a deterministic quantity \(c_{t,I}\) of magnitude \(\asymp N^{2\sigma -1}\) (the point being that the phase \(t \log n\) is close to constant in this range). By the Chernoff bound, we thus see that for any such \(t\), \(\sum _{n \in [N,2N]} a_n n^{it}\) will have size \(\gg N^{(2\sigma -1) + (2-2\sigma )/2} = N^\sigma \) with probability \(\gg 1\). By linearity of expectation, we thus see that with positive probability, a \(\gg 1\) fraction of integers \(t\) with \(t = o(N^{2-2\sigma })\) will have this property, giving the claim.

Finally, let \(\sigma =1/2\). In this case we just take each \(a_n\) to be a random sign, then by the Chernoff bound one has for each \(t\) that \(|\sum _{n \in [N,2N]} a_n n^{it}| \asymp N^{1/2}\) with positive probability, which by linearity of expectation as before gives the lower bound \(\mathrm{LV}(\sigma ,\tau ) \geq \tau \), while the upper bound is trivial from Lemma 7.4(iii).

Clear from Lemma 7.4(ii).

Let \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) be a large value pattern with \(T = N^{k\tau +o(1)}\) and \(V = N^{\sigma +o(1)}\), Raising 1 the \(k^{\mathrm{th}}\) power, we conclude that

for all \(t \in W\), where \(b_n\) is the Dirichlet convolution of \(k\) copies of \(a_n\), and thus is bounded by \(N^{o(1)}\) thanks to divisor bounds. Subdividing \([N^k, 2^k N^k]\) into \(k\) intervals of the form \([N',2N']\) for \(N' \asymp N^k\) and applying Definition 7.2 (with \(N, T, V\) replaced by \(N', T, V^k\)) we conclude that

and the claim then follows.

Let \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) be a large value pattern with \(T = N^{\tau + o(1)}\), \(V = N^{\sigma + o(1)}\). Applying [ 149 , Theorem 9.4 ] (with \(N\), \(T\) replaced with \(2N\), \(2T\) respectively and taking \(a_n = 0\) for \(n {\lt} N\)) one has

from which the result follows.

Set \(\rho := \mathrm{LV}(\sigma ,\tau )\); we may assume without loss of generality that \(\rho \geq 0\). Then by Definition 7.2, we can find a large value pattern \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) with \(N {\gt} 1\) unbounded, \(T = N^{\tau + o(1)}\), \(V = N^{\sigma + o(1)}\), and \(|W| = N^{\rho +o(1)}\). From 1 we have

hence for some \(1\)-bounded coefficients \(c_t\)

We apply the Halász argument. Interchanging the summations and applying Cauchy–Schwarz, we conclude that

hence on squaring and using the triangle inequality

In the case \(|t - t'| \leq N^{1-\varepsilon }\) for any fixed \(\varepsilon {\gt}0\), one can use Lemma 4.4 to obtain the bound

The total contribution of this case can then be bounded by \(N^{1+o(1)} R = N^{1+\rho +o(1)}\), thanks to the \(1\)-separation. In the remaining cases \(|t - t'| \geq N^{1-o(1)}\), we use Definition 4.2 to see that

and thus

By hypothesis, the second term on the right-hand side is asymptotically smaller than the left-hand side, and so we obtain \(\rho \leq 2-2\sigma \) as required.

By Lemma 7.7 it suffices to prove the latter claim. From Lemma 5.3 one has \(\beta (1/\tau ') \tau ' \leq k \tau ' + (\ell -k)\) and so the condition 4 holds whenever

The claim follows.

Apply Corollary 7.11 with the pair \((k,\ell ) = (1/2,1/2)\) from Lemma 5.10.

Follows from [ 107 , Lemma 1 ] .

By Lemma 7.7 it suffices to show that \(\mathrm{LV}(\sigma ,\tau ) \leq 2-2\sigma \) for \(\tau \leq 11\sigma -8\). From the previous theorem, and setting \(\rho = \mathrm{LV}(\sigma ,\tau )\), we have either

or

The latter bound can be rearranged as

and thus

and the claim follows. (See also the arguments in the first paragraph of [ 107 , p. 226 ] .)

Let \((N,T,V,(a_n)_{n \in [N,2N]},J,W)\) be a large value pattern with \(N \geq 1\) be unbounded, \(T = N^{\tau + o(1)}\), \(V = N^{\sigma + o(1)}\), and \(|W| = N^{\rho +o(1)}\). By [ 106 , Lemma 6 ] we have

and thus

Since \(\sigma {\gt} 3/4\), we can delete the second term \(2\rho +1/2\) on the right-hand side. Solving for \(\rho \), we conclude that

and taking suprema in \(\rho \), we obtain the claim.

See [ 160 , (1.4) ] (setting \(V = N^{\sigma +o(1)}\), \(T = N^{\tau +o(1)}\), and \(G \leq N\)). We remark that this form is an optimized form of the inequality after (3.2) in Jutila’s paper, which in our notation would read that

whenever \(\mathrm{LV}(\sigma ,\tau ) \leq \rho \). The optimization follows from Lemma 7.7 and routine calculations.