Exponential sums and Weyl differencing

Use $e(t)=\exp(2\pi i t)$, so changing a phase by an integer changes nothing. A sum of $N$ such terms has magnitude at most $N$ by the triangle inequality. For a linear phase with nonintegral slope, the geometric-sum formula instead bounds the magnitude by the smaller of $N$ and a constant times the reciprocal distance of the slope from the nearest integer. Near an integer slope, adjacent vectors nearly align, and substantial cancellation cannot be expected merely because complex exponentials appear.

For a real polynomial of degree $d\geq2$ with leading coefficient $\alpha$, the difference $P(n+h)-P(n)$ has degree $d-1$ when the integer shift is nonzero, with leading coefficient $d\alpha h$. Expanding the squared magnitude of the original sum groups pairs of indices by their difference, producing sums with these lower-degree phases over overlapping intervals. Repeated differencing and Cauchy–Schwarz eventually reduce the problem to linear sums. Their dependence on products of shifts explains both divisor estimates and losses in the classical argument.

Fix an integer degree $d\geq2$ and a positive real $\varepsilon$. Let the real polynomial have leading coefficient $\alpha$, and suppose coprime integers $a,q$ satisfy $q\geq1$ and $|\alpha-a/q|\leq q^{-2}$. The classical Weyl inequality is the displayed bound, with a constant depending only on the degree and epsilon, uniformly in the lower coefficients. The rational approximation concerns the leading coefficient of the stated polynomial degree, not an arbitrary coefficient that happens to have a convenient denominator.

The exponent is $1/2^{d-1}$, reflecting the repeated squaring in the proof. A power saving follows when the denominator lies in a range such as $N^\delta\leq q\leq N^{d-\delta}$ for a fixed positive $\delta$ in a nonempty range, choosing epsilon small enough. Indeed each term inside the parentheses then decreases by a fixed power of $N$. If the denominator stays fixed, its reciprocal does not decrease, so this estimate alone does not imply a little-oh bound compared with $N$. The trivial bound may remain the better numerical estimate for small endpoints.

A rational polynomial phase can be periodic modulo one. A full-period sum can then be evaluated exactly, but repeating the same period repeats its residual vector as well. A square-root-sized complete sum over one modulus does not automatically become a square-root-sized sum over arbitrarily many repeated periods. This distinction prevents confusing a complete Gauss-sum identity with a uniform bound for every incomplete interval or endpoint. The arithmetic structure of the phase, not just visual oscillation, governs the answer.

Exponential sums enter counting through finite Fourier orthogonality: averaging $e(an/q)$ over a complete set of additive frequencies detects whether the integer is divisible by the modulus. Substituting polynomial values turns congruence counts into phase sums, and differencing can bound the contributions of nonzero frequencies. For sequences modulo one, corresponding cancellation criteria connect sums to equidistribution, but the relevant limits and all fixed nonzero integer frequencies must be checked. One attractive finite cancellation example is an illustration, not by itself an equidistribution theorem.