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Here is one little-known and one completely unknown result.

The little known result is the Mean Motion Theorem. This says that for all real numbers $\lambda_j$ and all complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).

This result was conjectured by Lagrange, coming from celestial mechanics, and was proved in full generality by the combined efforts of H. Weyl, P. Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave in 1945. It seems that the subject was forgotten after the 1940s.

The completely unknown result is a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that $$\phi(t)=mt+O(1).$$ This is due to Piers Bohl in 1909. I have never seen any reference on this stronger result, or any discussion of possible generalization to larger $n$.

Weyl, Wintner and Hartman refer to Bohl proving the $n=3$ case of their results, the first non-trivial case, but do not discuss the $O(1)$. Favorov's paper from 2008 has Bohl's paper in the reference list but also does not discuss the $O(1)$. In fact I have not seen ANY mention of a more precise error term than $o(t)$ in the literature. A number of papers GENERALIZE the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark. I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it is published in German. However there exists a Russian translation (which is difficult to obtain), so I post it here for the benefit of this community.

Remark 2. The only book I know which addresses the subject is Sternberg's 1969 book. (This book has a rare distinction: it is not reviewed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)

Weyl's 1938 paper is very well written, fortunately in English, and accessible to a non-specialist. If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary.

References in chronological order:

Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909).

Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938)

B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969.

Favorov, S. Yu., Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324. MR2424001

Here is one little-known and one completely unknown result.

The little known result is the Mean Motion Theorem. This says that for all real numbers $\lambda_j$ and all complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).

This result was conjectured by Lagrange, coming from celestial mechanics, and was proved in full generality by the combined efforts of H. Weyl, P. Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave in 1945. It seems that the subject was forgotten after the 1940s.

The completely unknown result is a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that $$\phi(t)=mt+O(1).$$ This is due to Piers Bohl in 1909. I have never seen any reference on this stronger result, or any discussion of possible generalization to larger $n$.

Weyl, Wintner and Hartman refer to Bohl proving the $n=3$ case of their results, the first non-trivial case, but do not discuss the $O(1)$. Favorov's paper from 2008 has Bohl's paper in the reference list but also does not discuss the $O(1)$. In fact I have not seen ANY mention of a more precise error term than $o(t)$ in the literature. A number of papers GENERALIZE the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark on references. The only book I know which addresses the subject is Sternberg's 1969 book. (This book has a rare distinction: it is not reviewed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)

Weyl's 1938 paper is very well written, fortunately in English, and accessible to a non-specialist. If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary. I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it was published in German.

References in chronological order:

Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909). (There is a Russian translation which is difficult to obtain, so I post it here for the benefit of this community.)

Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938)

B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969.

Favorov, S. Yu., Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324. MR2424001

The Mean Motion Theorem says the following: for all real numbers $\lambda_j$ and all complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).

The question comes from celestial mechanics. This result was conjectured by Lagrange, and proved in full generality by the combined efforts of H. Weyl, Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

It is not "completely unknown", I would say "little known". It seems that the subject was forgotten after the 1940s.

Sometimes it is mentioned that Piers Bohl proved the first non-trivial case $n=3$. But what is really "completely unknown" is that Bohl proved a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that $$\phi(t)=mt+O(1).$$ I have never seen any reference on this stronger result, or any discussion of possible generalization to larger $n$.

Ref. Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909).

The only recent paper which addresses the subject which I know is

MR2424001 Favorov, S. Yu. Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324.

It has Bohl's paper in the reference list but does not discuss the $O(1)$. Neither Weyl, Wintner and Hartman do. In fact I have not seen ANY mention of this $O(1)$ (or any other more precise error term than $o(t)$) in the literature. A number of papers GENERALIZE the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark. I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it is published in German. However there exists a Russian translation (which is difficult to obtain), so I post it here for the benefit of this community.

Remark 2. (I answer the question in the comments). The only book which addresses the subject that I know is

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969.  (This book has a rare distinction: it is not reviewed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)

Weyl's paper is very well written, fortunately in English, and accessible to a non-specialist: Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938).

If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary.

Here is one little-known and one completely unknown result.

The little known result is the Mean Motion Theorem. This says that for all real numbers $\lambda_j$ and all complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).

This result was conjectured by Lagrange, coming from celestial mechanics, and was proved in full generality by the combined efforts of H. Weyl, P. Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave in 1945. It seems that the subject was forgotten after the 1940s.

The completely unknown result is a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that $$\phi(t)=mt+O(1).$$ This is due to Piers Bohl in 1909. I have never seen any reference on this stronger result, or any discussion of possible generalization to larger $n$.

Weyl, Wintner and Hartman refer to Bohl proving the $n=3$ case of their results, the first non-trivial case, but do not discuss the $O(1)$. Favorov's paper from 2008 has Bohl's paper in the reference list but also does not discuss the $O(1)$. In fact I have not seen ANY mention of a more precise error term than $o(t)$ in the literature. A number of papers GENERALIZE the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark. I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it is published in German. However there exists a Russian translation (which is difficult to obtain), so I post it here for the benefit of this community.

Remark 2. The only book I know which addresses the subject is Sternberg's 1969 book.  (This book has a rare distinction: it is not reviewed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)

Weyl's 1938 paper is very well written, fortunately in English, and accessible to a non-specialist. If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary.

References in chronological order:

Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909).

Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938)

B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969.

Favorov, S. Yu., Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324. MR2424001

The Mean Motion Theorem says the following: for every real numbers $\lambda_j$ and every complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is no much loss in generality if one assumes for simplicity that the sum has no real zeros).

The question comes from celestial mechanics. This result was conjectured by Lagrange, and proved in full generality by the combined efforts of H. Weyl, Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

It is not "completely unknown", I would say "little known". It seems that the subject was forgotten after 1940s.

Sometimes it is mentioned that Pierce Bohl proved the first non-trivial case $n=3$. But what is really "completely unknown" is that Bohl proved for $n=3$ and under some additional conditions on $\lambda_j$ and $a_j$ a much stronger statement, namely that $$\phi(t)=mt+O(1).$$ I have never seen any reference on this stronger result, and any discussion of possible generalization to larger $n$.

Ref. Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909).

The only recent paper which addresses the subject which I know is

MR2424001 Favorov, S. Yu. Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324.

It has Bohl's paper in the reference list but does not discuss the $O(1)$. Neither Weyl, Wintner and Hartman do. In fact I have not seen ANY mentioning of this $O(1)$ (or any other more precise error term than $o(t)$) in the literature. There is a number of papers GENERALIZING the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark. I suspect that the reason of such neglect is that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it is published in German. However there exists a Russian translation (which is difficult to obtain), so I
 post it here for the benefit of this community.

Remark 2. (I answer the question in the comments). The only book which addresses the subject that I know is

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969. (This book has a rare distinction: it is not refereed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to the possibility of constructing a calendar:-)

Weyl's paper is very well written, fortunately in English, and accessible to a non-specialist: Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938).

If you can read German or Russian, Bohl's paper is also a good reading, it is completely elementary.

The Mean Motion Theorem says the following: for all real numbers $\lambda_j$ and all complex numbers $a_j$ the following limit exists: $$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$ where $t$ is real. (There is a natural way to define what happens to the $\arg$ at the zeros, but there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).

The question comes from celestial mechanics. This result was conjectured by Lagrange, and proved in full generality by the combined efforts of H. Weyl, Hartman and A. Wintner in the 1930s. The final result, without any restrictions on $\lambda_k,a_k$ is due to B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions. Acta Math. 77, (1945). 137–279.

It is not "completely unknown", I would say "little known". It seems that the subject was forgotten after the 1940s.

Sometimes it is mentioned that Piers Bohl proved the first non-trivial case $n=3$. But what is really "completely unknown" is that Bohl proved a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that $$\phi(t)=mt+O(1).$$ I have never seen any reference on this stronger result, or any discussion of possible generalization to larger $n$.

Ref. Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J. für Math. 135, 189-283 (1909).

The only recent paper which addresses the subject which I know is

MR2424001 Favorov, S. Yu. Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324.

It has Bohl's paper in the reference list but does not discuss the $O(1)$. Neither Weyl, Wintner and Hartman do. In fact I have not seen ANY mention of this $O(1)$ (or any other more precise error term than $o(t)$) in the literature. A number of papers GENERALIZE the mean motion theorem to infinite sums. But nobody addresses the improvement of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.

Remark. I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it is published in German. However there exists a Russian translation (which is difficult to obtain), so I  post it here for the benefit of this community.

Remark 2. (I answer the question in the comments). The only book which addresses the subject that I know is

S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969. (This book has a rare distinction: it is not reviewed in Mathscinet:-) The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)

Weyl's paper is very well written, fortunately in English, and accessible to a non-specialist: Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938).

If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary.