[proofplan] We first convert the small-fractional-parts count into a discrepancy estimate for the finite sequence $x_1,\dots,x_N$ modulo $1$. The discrepancy bound for the two intervals near $0$ gives a main term equal to their total length, namely $2\delta$, and an error term $2D_N$. We then insert the Erdős-Turán discrepancy inequality, which controls $D_N$ by the first $H$ exponential sums. Multiplying out the constants gives the stated estimate. [/proofplan] [step:Bound the small-fractional-parts count by discrepancy] Let $D_N(x_1,\dots,x_N)$ denote the discrepancy of the finite sequence $x_1,\dots,x_N$ modulo $1$. Since $N\in\mathbb N$, $x_1,\dots,x_N\in\mathbb R$, and $0<\delta\le 1/2$, the hypotheses of [citetheorem:9095] apply to the sequence $x_1,\dots,x_N$ and the radius $\delta$. Therefore \begin{align*} \#\{1\le n\le N:\|x_n\|_{\mathbb R/\mathbb Z}<\delta\}\le 2\delta N+2N D_N(x_1,\dots,x_N). \end{align*} [guided] Let $D_N(x_1,\dots,x_N)$ denote the discrepancy of the finite sequence $x_1,\dots,x_N$ modulo $1$. The point of introducing discrepancy is that the condition $\|x_n\|_{\mathbb R/\mathbb Z}<\delta$ means that the fractional part of $x_n$ lies near $0$ in the circle $\mathbb R/\mathbb Z$. More precisely, the relevant region is contained in the union of the two endpoint intervals \begin{align*} [0,\delta)\cup[1-\delta,1). \end{align*} The total length of these intervals is $2\delta$ when $0<\delta\le 1/2$, and the discrepancy measures how far the actual number of fractional parts in such intervals can deviate from the expected number. The theorem [citetheorem:9095] applies because its hypotheses are exactly that $x_1,\dots,x_N\in\mathbb R$ and $0<\delta\le 1/2$. It gives \begin{align*} \#\{1\le n\le N:\|x_n\|_{\mathbb R/\mathbb Z}<\delta\}\le 2\delta N+2N D_N(x_1,\dots,x_N). \end{align*} The strict inequality in $\|x_n\|_{\mathbb R/\mathbb Z}<\delta$ causes no endpoint difficulty here because the cited result is already stated for this strict small-fractional-parts count and gives an upper bound. [/guided] [/step] [step:Insert the Erdős-Turán discrepancy inequality] Since $H\in\mathbb N$, the hypotheses of [citetheorem:9093] apply to the [real numbers](/page/Real%20Numbers) $x_1,\dots,x_N$ with truncation parameter $H$. Hence \begin{align*} D_N(x_1,\dots,x_N)\le \frac{3}{H+1}+\frac{3}{N}\sum_{h=1}^{H}\frac{1}{h}\left|\sum_{1\le n\le N}e(hx_n)\right|. \end{align*} Substituting this estimate into the previous step gives \begin{align*} \#\{1\le n\le N:\|x_n\|_{\mathbb R/\mathbb Z}<\delta\}\le 2\delta N+2N\left(\frac{3}{H+1}+\frac{3}{N}\sum_{h=1}^{H}\frac{1}{h}\left|\sum_{1\le n\le N}e(hx_n)\right|\right). \end{align*} Distributing the factor $2N$ yields \begin{align*} \#\{1\le n\le N:\|x_n\|_{\mathbb R/\mathbb Z}<\delta\}\le 2\delta N+\frac{6N}{H+1}+6\sum_{h=1}^{H}\frac{1}{h}\left|\sum_{1\le n\le N}e(hx_n)\right|. \end{align*} This is the desired estimate. [/step]