Covariance of the number of solutions

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.