On pair correlations and Hausdorff dimension (Q836114)

Consider a system of generalized linear forms \(L_i(x)(k)=\sum_{j=1}^{d_i} g_{(i,j),k}(x_{(i,j)})\), \(i=1,\dots,L\), where \(x=(x_{(1,1)},\dots,x_{(1,d_1)},\dots,x_{(l,1)},\dots,x_{(l,d_l)})\) and for every \((i,j)\) the sequence of functions \(g_{(i,j),k}\) satisfies the following conditions: (1) for \(m\neq n\) the function \(x\mapsto g_{(i,j),m}^{(p_{(i,j)})}(x)-g_{(i,j),n}^{(p_{(i,j)})}(x)\) is monotone, and (2) for \(m\neq n\) we have \(|g_{(i,j),m}^{(p_{(i,j)})}(x)-g_{(i,j),n}^{(p_{(i,j)})}(x)|\geq K_{(i,j)}>0\) for every \(x\). Here, \(g^{(p)}\) denotes the \(p\)-th derivative of \(g\). Let \[ X_k(x)=(\{L_1(x)(k)\},\dots,\{L_l(x)(k)\})\in{\mathbb T}^l, \] where \(\{x\}\) denotes the fractional part and \({\mathbb T}^l\) is the \(l\)-torus. For a rectangle \(R=I_1\times\dots\times I_l\subset{\mathbb T}^l\) and \(N\geq 1\) define a pair correlation function \[ V_N(R)(x)=\sum_{1\leq n\neq m\leq N} \chi_R(X_n(x)-X_m(x)) \] and a discrepancy \[ \Delta_N(x)=\sup_{R\subset{\mathbb T}^l}(V_N(R)(x)-N(N-1)\lambda(R)), \] where \(\chi\) is the characteristic function and \(\lambda\) is the Lebesgue measure. The author gives estimations of \(\Delta_N(x)\) and of the Hausdorff dimension of some exceptional sets. Let \(p=\max p_{(i,j)}>1\) and \(\epsilon>0\). Then for Lebesgue almost every \(x\) \[ \Delta_N(x)=o(N^{2-1/p}(\log N)^{l+\epsilon}). \] Suppose that for every \((i,j)\) the sequence of functions \(g_{(i,j),k}\) satisfies in addition (3) \(\sup_x|g_{(i,j),k}^{p_{(i,j)}}(x)|\leq C_{(i,j)}k^{q_{(i,j)}}\) for some \(C_{(i,j)}>0\) and \(q_{(i,j)}>1\), and (4) for \(m\neq n\) the function \(x\mapsto g_{(i,j),m}^{(1)}(x)-g_{(i,j),n}^{(1)}(x)\) has finitely many intervals of monotonicity. Let \(p=\max p_{(i,j)}>1\), \(q=\max q_{(i,j)}\) and \(d=d_1+\dots+d_l\). Then for every \(t\in(0,\frac{1}{p-1})\) \[ \dim \{x|\limsup_{N\to\infty} N^t\delta_N(x)>0\}\leq d-\frac{\frac{1}{p}-t(1-\frac{1}{p})}{q+t}. \]