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Mark Lewko
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This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$$S(K) \ll K^{1/2 +1/6} = K^{2/3}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6} = K^{2/3}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.

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Mark Lewko
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This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ integersnatural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is.

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ integers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is. (Update: the paper of Konyagin cites the problem to Bohr's paper [3])

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ natural numbers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.

[3] H. Bohr, A study of the uniform convergence of Dirichlet series and its connection with a problem concerning ordinary polynomails, Fys. Sall. Lund Forth 21 (1951) 103-118.

Source Link
Mark Lewko
  • 13k
  • 1
  • 55
  • 87

This is sometimes referred to as the "sine problem" in parallel with the much better known "cosine problem" of Chowla. I've seen the problem attributed to Bohr in connection with a problem about Dirichlet Series, but I've never seen a reference for that and I am not aware of what the application is.

To state what is known, define

$S(K):= \min_{\{n_k\}_{k\in[K]}} \max_{\theta} |\sum_{1\leq k \leq K} \sin(n_k\theta) |$

where $\{n_k\}$ is an arbitrary set of $K$ integers. Konyagin [1] has shown that:

$S(K) \gg K^{1/2} \left( \frac{\log K}{ \log \log K}\right)^{1/2}$

which gives a negative answer to the question as posed. In the other direction, Bourgain [2] constructed examples showing that $S(K) \ll K^{1/2 +1/6}.$

[1] S. V. Konyagin, Estimates of maxima of sine sums, East J. Approx. 3 (1997), no. 1, 63–7

[2] J. Bourgain, Sur les sommes de sinus, Harmonic analysis: study group on translationinvariant Banach spaces, Exp. No. 3, 9 pp., Publ. Math. Orsay 83, 1, Univ. Paris XI, Orsay, 1983.