MULTIDIMENSIONAL HECKE THEOREM ON THE DISTRIBUTION OF FRACTIONAL PARTS

作者：Zhuravlev V G^*

来源：St Petersburg Mathematical Journal, 2013, 24(1): 71-97.

• Hecke theorem
• fractional parts distribution
• mean values of deviation functions
• bounded remainder sets on the torus

Hecke's theorem on the distribution of fractional parts on the unit circle is generalized to the tori T-D = R-D/L of arbitrary dimension D. It is proved that vertical bar delta(k)(i)vertical bar <= c(k) n for i = 0, 1, 2, ..., where delta(k)(i) = r(k)(i) - ia(k) is the deviation of the number r(k)(i) of returns in i steps into T-k(D) subset of T-D for the points of an S-beta-orbit from its mean value a(k) = vol(T-k(D))/ vol(T-D), where vol(T-k(D)) and vol(T-D) denote the volumes of the tile T-k(D) and of the torus T-D. The tiles T-k(D) in question have the following property: for the torus T-D there exists a development T-D subset of R-D such that a shift S-alpha of the torus T-D is equivalent to some exchange transformation of the corresponding tiles T-k(D) in a partition of the development T-D = T-0(D) coproduct T-1(D) coproduct ... T-D(D). The torus shift vectors S-alpha, S-beta satisfy the condition alpha n beta mod L, where n is any natural number, and the constants c(k) in the inequalities are expressed in terms of the diameter of the development T-D.

• 出版日期2013-2