On the theorem of Davenport and generalized Dedekind sums (Q344113)

The theorem of \textit{H. Davenport} [Mathematika 3, 131--135 (1956; Zbl 0073.03402)] states the following: for a badly approximable number \(\alpha\) the set \(A:=A(n,\alpha)=\{(\{\pm k\alpha\},\frac kn):1\leq k\leq n\}\subset [0,1]^2\) satisfies \(\int_0^1\int_0^1(S_A(x,y)-|A|xy)^2dxdy<C(\alpha) \log |A|\), where \(S_A(x,y)=|A\cap ([0,x)\times [0,y))|\). The author obtains more precise and extended results. Let \(\alpha\) be an irrational number with the continued fraction representation \(\alpha=[a_0, a_1, a_2, \ldots]\) and let \(p_k/q_k=[a_0, a_1, a_2, \ldots, a_{k-1}]\). The author shows (Theorem 1) that if the irrational number \(\alpha\) satisfies \(a_k=O(k^d)\) for some \(d\geq 0\), then the set \(A=A(n,\alpha)\) satisfies NEWLINE\[NEWLINE\int_0^1\int_0^1(S_A(x,y)-|A|xy)^2dxdy=\sum_{m=1}^n\frac{1}{4\pi^4m^2\|m\alpha\|^2}+ O(\log^{2d}n\log\log n).NEWLINE\]NEWLINE \textit{J. Beck} [Probabilistic Diophantine approximation. Randomness in lattice point counting. Cham: Springer (2014; Zbl 1304.11003)] showed that if \(\alpha\) is a quadratic irrational, then the Diophantine sum \(\sum_{m=1}^n\frac{1}{4\pi^4m^2\|m\alpha\|^2}=c(\alpha)\log n+O(1)\). The author offers ``a way to estimate the Diophantine sum in terms of the continued fraction representation of \(\alpha\)''(Theorem 3). By Theorem 1 and the result of Beck [loc. cit.], the author obtains the following result. If \(\alpha\) is a quadratic irrational, then NEWLINE\[NEWLINE\int_0^1\int_0^1(S_A(x,y)-|A|xy)^2dxdy=c(\alpha)\log n+O(\log\log n).NEWLINE\]NEWLINE It is also shown that if \(a_k=O(k^d)\) for some \(d>0\) and \(q_{\ell}\leq n<q_{\ell+1}\), then NEWLINE\[NEWLINE \int_0^1\int_0^1(S_A(x,y)-|A|xy)^2dxdy=\frac{1}{360}\sum_{k=1}^{\ell}a_k^2+ O(\log^{d+1}n+\log^{2d}n\log\log n).NEWLINE\]NEWLINE Moreover, the author proves that a ``generalized Dedekind sum can be approximated by a Diophantine sum similar to the one in Theorem 1 with a rational number \(\alpha=\frac ab\)'' (Theorem 2).