Probability estimates related to Korobov’s number-theoretical quadrature formulas

Let a 1 , … , a s a_1, , a_s be integers and let N N be a positive integer. Korobov (1959) and Hlawka (1962) proposed to use the points x ( k ) = ( { a 1 k / N } , … , { a 1 k / N } ) x⁽ᵏ⁾ = (\{a_1 k/N\}, , \{a_1 k/N\}) , k = 1 k=1 , …, N N , as nodes of multidimensional quadrature formulas. The paper presents some new probability estimates related to the discrepancy of the sequence K N ( a ) = { x ( 1 ) , … , x ( N ) } K_N(a)=\{x⁽¹⁾, , x(N)\} and the error of the Korobov–Hlawka number-theoretical quadrature formulas.