Uniform distribution and arbitrary Poincaré functions (Q2702774)

In this paper the author presents quantitative versions of two Poincaré results, the first deals with the motion of planets and the second involving roulettes. A common feature are sequences having limit laws different from uniform distribution and which are transformed (by relating operations) to nearly uniformly distributed sequences. The paper is divided into 7 sections and ends with comments. NEWLINENEWLINENEWLINESection 1 is devoted to the transformation of a two-dimensional sequence \((a_n,b_n)\), \(n=1,2,\dots \) to the one-dimensional one \(a_nt+b_n\bmod 1\). The author interpretes \(a_n\) as an angular velocity and \(b_n\) as a starting angle in the time \(t=0\) of the orbit of the \(n\)-th planet \(P_n\). (All angles are measured on the circle with the unit length.) If \((a_n,b_n)\) is distributed with density \(\Phi (a,b)\), then \(a_nt+b_n\bmod 1\) is uniformly distributed as \(t\to \infty \). Precisely, let \((a_n,b_n)\) be points in the interval \(K=[u,u+v]\times [0,1]\) and let \(\Phi (a,b)\) be a density defined on \(K\), i.e., \(\Phi (a,b)\geq 0\) and \(\iint_K\Phi (a,b)da db=1\). Define the extremal discrepancy \(\widetilde {D}_N\) of the sequence \((a_n,b_n)\) related to \(\Phi \) as NEWLINE\[NEWLINE\widetilde {D}_N=\sup_{J\subset K} \left |\frac {1}{N}\sum_{n=1}^Nc_J((a_n,b_n)) -\iint_J\Phi (a,b) da db\right |,NEWLINE\]NEWLINE where \(J\) are intervals and \(c_J(x,y)\) is the characteristic function of \(J\). Assuming that the partial derivatives \(\frac {\partial \Phi }{\partial a}\) and \(\frac {\partial \Phi }{\partial b}\) are bounded on \(K\), the author proves that for every \(t>(D_N((a_n,b_n)))^{-1/4}\) and every \(M>0\), the one-dimensional sequence \(a_nt+b_n\bmod 1\) has the classical extremal discrepancy NEWLINE\[NEWLINED_N\leq C_1(M^{-1}+C_2t^{-1}+\widetilde {D}_NtM^2),\tag{*}NEWLINE\]NEWLINE where \(C_1>0\) is an absolute constant and \(C_2>0\) depends on \(\Phi \).NEWLINENEWLINENEWLINESection 2 presents a transformation of a one-dimensional sequence \(\varphi_n\), \(n=1,2,\dots \) in \([0,1]\) to a discrete one 0-1 sequence by using a roulette. If \(\varphi_n\) is distributed with a density \(\rho (\varphi)\), then the resulting sequence is uniformly distributed. Precisely, let \(J_{kr}=\left [\frac {k}{s}+\frac {r}{2s}, \frac {k}{s}+\frac {r+1}{2s}\right ]\), \(r=0,1\), \(k=1,\dots ,s\), be a two-colouring decomposition of \([0,1]\) with two colours \(0\) and \(1\). The author identifies a roulette \(R\) with the couple of \(\rho (\varphi)\) and \(J_{kr}\), where the density \(\rho (\varphi)\) characterizes a rotation of the roulette. Let \(D^\rho_N\) denote the extremal discrepancy of \(\varphi_n\) with respect to \(\rho (\varphi)\). Put \(S^s_{r,N}=\frac {1}{N}\sum_{n=1}^N\sum_{k=1}^sc_{J_{kr}}(\varphi_n)\). Assuming a Lipschitz condition \(|\rho (\varphi)-\rho (\varphi ')|\leq \alpha |\varphi -\varphi '|\), the author proves NEWLINE\[NEWLINE|S^s_{j,N}-(1/2)|\leq (\alpha +2)(D^\rho_N)^{1/2}, \tag{**} NEWLINE\]NEWLINE for \(j=0,1\) and \(s=[(D_N(\varphi_n))^{-1/2}]+1\). NEWLINENEWLINENEWLINESection 3 covers \(m\)-coloured roulette and the author gives an analogue of (**). Section 4 deals with a composition of roulettes and the author proves a multidimensional analogue of (**). Section 5 includes a multidimensional sequence \((a_n,b_n)\) having density \(\Phi (a,b)=\Phi_1(a^1,b^1)\Phi_2(a^2,b^2)\dots \) and for which \(a_nt+b_n\) can be used for rotations of roulettes as \(\varphi_n=a_nt+b_n\bmod 1\). Again, the author presents an analogy of (*). NEWLINENEWLINENEWLINESection 6 contains a four-dimensional sequence \((u_n,v_n,a_n,b_n)\), \(n=1,2,\dots \) in \([0,\delta ]\times [0,1]\times [0,\delta ]\times [0,1]\) having distribution with density \(\rho (u,v,a,b)=\rho_1(u)\rho_2(v)\rho_3(a)\rho_4(b)\) and which is transformed to the two-dimensional sequence \(((u_n/\delta)\Phi (\mu ,t)+(a_n/\delta),v_n\Phi (\mu ,t)+b_n)\bmod 1\), where \(\Phi (\mu ,t)=(1-e^{-\mu t})/\mu \). For its discrepancy the author gives NEWLINE\[NEWLINED_N\leq C((D^\rho_N)/\mu^2)^{1/5}+\delta^2\mu^2),NEWLINE\]NEWLINE where the extremal discrepancy \(D^\rho_N\) of \((u_n,v_n,a_n,b_n)\) is related to \(\rho \). Here \(x=u_n\Phi (\mu ,t)+a_n\) and \(\omega =v_n\Phi (\mu ,t)+b_n\) solve the system of differential equations \(\dot {x}=p\), \(\dot \varphi =\omega \), \(\dot {p}=-\mu p\) and \(\dot \omega =-\mu \omega \) with a friction \(\mu >0\). NEWLINENEWLINENEWLINESection 7 is the major section of this paper. The author investigates a system of differential equations \(\dot {x}_j=f_j(p_j)\), \(\dot {p}_j=-g_j(x_j)\), \(j=1,\dots ,s\) which has implicit solutions \(G_j(x_j)+F_j(p_j)=E_j\) and in explicit form \(x_j=X_j(t,\xi ,\pi)\), \(p_j=P_j(t,\xi ,\pi)\), with starting condition \((x,p)=(\xi ,\pi)\), \(\xi =(\xi_1,\dots ,\xi_s)\), \(\pi =(\pi_1,\dots ,\pi_s)\). Let \(\pi^n=(\pi^n_1,\dots ,\pi^n_s)\) be a sequence with density \(\rho (\pi)=\rho_1(\pi_1)\dots \rho_s(\pi_s)\) and discrepancy \(D^\rho_N\). The author transforms \(\pi^n\) to the sequence \(F(\pi^n)=F_1(\pi^n_1),\dots ,F_s(\pi^n_s)\) (here \(F_j\) has some normed form) and for its discrepancy \(D_N\) he gives a bound depending on \(D^\rho_N\). Then, using \(G_j(\xi^n_j)+F_j(\pi^n_j)=E_j\) (in some normed form) the author constructs a new sequence \(\xi^n=(\xi^n_1,\dots ,\xi^n_s)\) with density \(g(\xi)=g_1(\xi_1)\dots g_s(\xi_s)\) and discrepancy \(D^g_N\leq 2D_N\). Now, let \((x^k,p^n)\), \(k,n=1,2,\dots \) be a double sequence of the solution of the system related to the starting sequence \((\xi^k,\pi^n)\) and \(X(x,p)\) be a function with bounded variation. The author proves that the sum \(N^{-2}\sum_{k,n=1}^NX(x^k,p^n)\) can be computed as \(N^{-2}\sum_{k,n=1}^NX(\xi^k,\pi^n)\) with error term \(C\text{ Var }X(D^g_N+D^\rho_N)\). NEWLINENEWLINENEWLINEAll discrepancy bounds in the paper are obtained by combining the classical multidimensional Erdős-Turán-Koksma inequality and the Koksma-Hlawka inequality [\textit{M. Drmota} and \textit{R. F. Tichy}, Sequences, Discrepancies and Applications, Springer-Verlag, Berlin (1997; Zbl 0877.11043)].