Difference between revisions of "Covariance of the number of solutions"
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A concept in the [[Dispersion method|dispersion method]] introduced for the comparison of the number of solutions of the equations | A concept in the [[Dispersion method|dispersion method]] introduced for the comparison of the number of solutions of the equations | ||
| − | + | $$ \tag{1 } | |
| + | n = \phi + D ^ { \prime } \nu | ||
| + | $$ | ||
and | and | ||
| − | + | $$ \tag{2 } | |
| + | n = \psi + D ^ { \prime } \nu , | ||
| + | $$ | ||
| − | where | + | where $ \phi $ |
| + | and $ \psi $ | ||
| + | belong to certain sequences of positive integers, $ D ^ { \prime } $ | ||
| + | runs through some given system of intervals on the real axis | ||
| − | + | $$ | |
| + | ( D) = [ D _ {1} , D _ {1} + D _ {2} ] , | ||
| + | $$ | ||
| − | and | + | and $ \nu $ |
| + | runs through a system of intervals on the real axis | ||
| − | + | $$ | |
| + | ( \nu ) = [ \nu _ {0} , \nu _ {0} + \nu _ {0} ^ \prime ] . | ||
| + | $$ | ||
Let | Let | ||
| − | + | $$ | |
| + | U _ {1} ( m) = \ | ||
| + | \sum _ {\phi = m } 1 ,\ \ | ||
| + | U _ {2} ( m) = \ | ||
| + | \sum _ {\psi = m } 1 , | ||
| + | $$ | ||
then the dispersion of the difference of the solutions of (1) and (2) is | then the dispersion of the difference of the solutions of (1) and (2) is | ||
| − | + | $$ | |
| + | V ^ { \prime } = \ | ||
| + | \sum _ {D ^ { \prime } \in ( D) } | ||
| + | \left ( {\sum _ {1} } \prime - {\sum _ {2} } \prime \right ) ^ {2} , | ||
| + | $$ | ||
where | where | ||
| − | + | $$ | |
| + | {\sum _ {1} } \prime = \ | ||
| + | \sum _ {\nu \in ( \nu ) } | ||
| + | U _ {1} ( n - D ^ { \prime } \nu ) ,\ \ | ||
| + | {\sum _ {2} } \prime = \ | ||
| + | \sum _ {\nu \in ( \nu ) } | ||
| + | U _ {2} ( n - D ^ { \prime } \nu ) . | ||
| + | $$ | ||
| − | Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over | + | Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over $ D ^ { \prime } $ |
| + | to all of $ D $ | ||
| + | in $ ( D) $. | ||
| + | This can only increase the dispersion; thus | ||
| − | + | $$ | |
| + | V ^ { \prime } \leq V = \ | ||
| + | V _ {1} - 2 V _ {2} + V _ {3} , | ||
| + | $$ | ||
where | where | ||
| − | + | $$ | |
| + | V _ {1} = \ | ||
| + | \sum _ {D \in ( D) } | ||
| + | ( \sum _ {1} ) ^ {2} , | ||
| + | $$ | ||
| − | + | $$ | |
| + | V _ {3} = \sum _ {D \in ( D) } ( \sum _ {2} ) ^ {2} , | ||
| + | $$ | ||
| − | + | $$ | |
| + | V _ {2} = \sum _ {D \in ( D) } ( \sum _ {1} \sum _ {2} ) ; | ||
| + | $$ | ||
here | here | ||
| − | + | $$ | |
| + | \sum _ {1} = \ | ||
| + | \sum _ {\nu \in ( \nu ) } | ||
| + | U _ {1} ( n - D \nu ) , | ||
| + | $$ | ||
| − | + | $$ | |
| + | \sum _ {2} = \sum _ {\nu \in ( \nu ) } U _ {2} ( n - D \nu ) . | ||
| + | $$ | ||
| − | In analogy with probability-theoretic concepts, | + | In analogy with probability-theoretic concepts, $ V _ {2} $ |
| + | is called the covariance of the number of solutions of (1) and (2). An asymptotic estimate of $ V _ {1} $, | ||
| + | $ V _ {3} $ | ||
| + | and the covariance $ V _ {2} $ | ||
| + | shows that the dispersion $ V ^ { \prime } $ | ||
| + | is relatively small, and this is essential in considering additive problems that lead to equations (1) and (2). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
See also [[Circle method|Circle method]]. | See also [[Circle method|Circle method]]. | ||
Latest revision as of 17:31, 5 June 2020
A concept in the dispersion method introduced for the comparison of the number of solutions of the equations
$$ \tag{1 } n = \phi + D ^ { \prime } \nu $$
and
$$ \tag{2 } n = \psi + D ^ { \prime } \nu , $$
where $ \phi $ and $ \psi $ belong to certain sequences of positive integers, $ D ^ { \prime } $ runs through some given system of intervals on the real axis
$$ ( D) = [ D _ {1} , D _ {1} + D _ {2} ] , $$
and $ \nu $ runs through a system of intervals on the real axis
$$ ( \nu ) = [ \nu _ {0} , \nu _ {0} + \nu _ {0} ^ \prime ] . $$
Let
$$ U _ {1} ( m) = \ \sum _ {\phi = m } 1 ,\ \ U _ {2} ( m) = \ \sum _ {\psi = m } 1 , $$
then the dispersion of the difference of the solutions of (1) and (2) is
$$ V ^ { \prime } = \ \sum _ {D ^ { \prime } \in ( D) } \left ( {\sum _ {1} } \prime - {\sum _ {2} } \prime \right ) ^ {2} , $$
where
$$ {\sum _ {1} } \prime = \ \sum _ {\nu \in ( \nu ) } U _ {1} ( n - D ^ { \prime } \nu ) ,\ \ {\sum _ {2} } \prime = \ \sum _ {\nu \in ( \nu ) } U _ {2} ( n - D ^ { \prime } \nu ) . $$
Applying an idea of I.M. Vinogradov on smoothing double sums, one can extend the summation over $ D ^ { \prime } $ to all of $ D $ in $ ( D) $. This can only increase the dispersion; thus
$$ V ^ { \prime } \leq V = \ V _ {1} - 2 V _ {2} + V _ {3} , $$
where
$$ V _ {1} = \ \sum _ {D \in ( D) } ( \sum _ {1} ) ^ {2} , $$
$$ V _ {3} = \sum _ {D \in ( D) } ( \sum _ {2} ) ^ {2} , $$
$$ V _ {2} = \sum _ {D \in ( D) } ( \sum _ {1} \sum _ {2} ) ; $$
here
$$ \sum _ {1} = \ \sum _ {\nu \in ( \nu ) } U _ {1} ( n - D \nu ) , $$
$$ \sum _ {2} = \sum _ {\nu \in ( \nu ) } U _ {2} ( n - D \nu ) . $$
In analogy with probability-theoretic concepts, $ V _ {2} $ is called the covariance of the number of solutions of (1) and (2). An asymptotic estimate of $ V _ {1} $, $ V _ {3} $ and the covariance $ V _ {2} $ shows that the dispersion $ V ^ { \prime } $ is relatively small, and this is essential in considering additive problems that lead to equations (1) and (2).
References
| [1] | Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian) |
Comments
See also Circle method.
How to Cite This Entry:
Covariance of the number of solutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariance_of_the_number_of_solutions&oldid=46541
Covariance of the number of solutions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariance_of_the_number_of_solutions&oldid=46541
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article