CHAP. II] REAL LINEAR FORMS, APPROXIMATION. 97 



S. Kakeya 275 proved the theorem (Minkowski, 264 p. 108) that if ai, , a n 

 are real there exist integers Xi, , x n , z such that Xi a\z, ,# a n z 

 are as small numerically as we please. He proved that these forms approach 

 indefinitely any real numbers. He 274 gave a generalization to any linear 

 functions. 



R. Remak 2740 proved arithmetically Minkowski' s 268 first theorem. 



H. Weber and J. Wellstein 2746 gave a new arithmetical proof of Min- 

 kowski's 264 initial theorem for both real and imaginary linear forms. 



H. F. Blichfeldt 275 proved a result which in Minkowski's notations 

 becomes 



For small values of n this limit is higher than Minkowski's 264 (p. 122), but 

 for large n's it is smaller. Given the positive numbers ai, , a n -i and 

 any positive number b < |, we can find integers Xi, , Xn-i, Z such 

 that the n 1 differences | XJZ at \ are ^ 26 and 



y (n - 



n 



I'+O 



Except for n = 2, this approximation is closer than that obtained by 

 Hermite, 259 Kronecker, and Minkowski. 264 



G. H. Hardy and J. E. Littlewood 276 proved that if 0i, , 8 m are ir- 

 rational and connected by no linear relation with integral coefficients not 

 all zero, and if an, -, akm are any numbers such that ^ a,-/ < 1, there 

 exists a sequence of positive integers HI, nz, - such that the fractional 

 part of nldp approaches a tp as r increases, for I = 1, , k; p = 1, , m 

 (the case k = 1 being due to Kronecker 262 ). Given X, there is therefore a 

 function 3> of k, m, X and the 0's and a's such that the difference between 

 the fractional part (n l 6 p ) of n l d p and ai p is numerically < 1/X for some n < $>. 

 When the 0's are given, a $ can be found independent of the as. When 

 all the a's are zero, a $ can be found independent of the 0's. An upper 

 bound for 3>, in this last case, was later given by H. T. J. Norton. 277 H. 

 Weyl 277a went further by showing that the numbers (n l d p ) are " uniformly 

 distributed ' ' throughout the unit cube ^= XI P ^= 1 in space of km di- 

 mensions [i.e., if we associate with n the point whose km coordinates are 

 XI P = (n l 0p) and denote by n v the number of the first n points which lie 

 inside an assigned part of the cube, of volume V, then n v ~ nV when 

 n -> 



273 Science Reports Tohoku University, 2, 1913, 33-54. 



274 T6hoku Math. Jour., 4, 1913-4, 120-131. 



2740 Jour, fur Math., 142, 1913, 278-82. 



2745 Math. Annalen, 73, 1913, 275-85. 



275 Trans. Amer. Math. Soc., 15, 1914, 227-235. 



8 



