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For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680, where you can also find the precise asymptotics of $K_{it}$ in the transitional range.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{*}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (*) in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) \|\phi\|_2. $$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}\|\phi\|_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).

For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680, where you can also find the precise asymptotics of $K_{it}$ in the transitional range.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{1}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (1) above in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) {\|\phi\|}_2. \tag{2}$$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}{\|\phi\|}_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).

Added. For a generalization of (2) to spherical Hecke-Maass forms over an arbitrary number field, see Lemma 9 in this paper. Note that $\phi$ in that lemma satisfies ${\|\phi\|}_2=1$. Note also that (1) above was (*) in an earlier version of this post.

For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{*}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (*) in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) \|\phi\|_2. $$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}\|\phi\|_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).

For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680, where you can also find the precise asymptotics of $K_{it}$ in the transitional range.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{*}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (*) in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) \|\phi\|_2. $$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}\|\phi\|_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).

For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{*}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (*) in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) \|\phi\|_2. $$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $|\phi(x+ir/(2\pi))|$ is about $r^{1/6}\|\phi\|_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).

For a published account of the corrected proof, see Section 10 in Blomer-Holowinsky: Bounding sup-norms of cusp forms of large level, Invent. Math. 179 (2010), 645-681. See especially pages 679-680.

Actually, a few years ago, a colleague of mine asked me about the same thing. What follows is my email response in slightly edited form (I am too busy to re-read carefully):

There are several problems with Lemma A.1 and its proof. The problems arose from not using correctly the asymptotics of $K_{ir}(y)$ in the transitional range $y\approx r$. For $|y-r|<r^{1/3}$ the function is about $e^{-\pi r/2} r^{-1/3}$ times a phase, while for $|y-r|>r^{1/3}$ it is about $e^{-\pi r/2} |y^2-r^2|^{-1/4}$ times a phase, with the additional remark that for $y$ large the exponential decay kicks in. As a consequence:

1. The display below (A.2) is wrong: $r^{-1}$ should be lowered to $r^{-4/3}$. This is not a big problem, since if we integrate from $r/2$ (instead from $r$), then the bound is OK (I learned these things from papers by Strömbergsson). In other words, in the line before (A.3) one should set $N=r/(4\pi Y)$. Actually (A.3) is too weak for this purpose, as explained below. Instead the following should be used, which is known by deeper methods: $$ \sum_{0<|n|<N} |\rho(n)|^2 \ll e^{\pi r} r^\epsilon N. \tag{*}$$

2. The last display on p.316 is wrong. In fact for fixed $b\geq 0$ one has $$ \sum_{n\neq 0} |n|^b |K_{ir}(2\pi|n|y)|^2 \ll e^{-\pi r} (r/y)^b (r^{-2/3}+1/y), $$ and this is best possible for $2\pi y < r+O(1)$, e.g. the first term in the parantheses comes from $2\pi|n|y$ very close to $r$. This shows that the last display on p.316 should read $$ |\phi(z)|^2 \ll Y^{-2} (Y+r) (y^{-1}r+Y)^2 (yr^{-2/3}+1). $$ For the choice $Y=r/(2\pi y)$ this gives $$ |\phi(z)|^2 \ll (Y+r) (yr^{-2/3}+1), $$ which is too weak eventually. Using (*) in place of (A.3), one gets $$ |\phi(z)|^2 \ll r^\epsilon Y (yr^{-2/3}+1), $$ i.e., $$ \phi(z) \ll r^\epsilon (r^{1/6} + (r/y)^{1/2}) \|\phi\|_2. $$ This is a bit weaker than Lemma A.1. In fact this is the correct version of Lemma A.1 since $\max_{0\leq x\leq 1}|\phi(x+ir/(2\pi))|$ is about $r^{1/6}\|\phi\|_2$.

Much the same correction is explained on pp.38-40 of Sarnak's letter to Morawetz.

A bit more detail is given in the Blomer-Holowinsky paper, on p.679. Actually, the bound on top of p.680 is only justified for $t>1$ and $y>1$, but it is valid for all $y>0$ and all forms (including forms violating the Selberg conjecture).