The study of exponential sums occupies a central position in analytic number theory due to their deep connections with problems in Diophantine approximation, the distribution of prime num-bers, and Waring–type additive questions. Among these, G. Weyl’s trigonometric sums play a particularly important role, providing powerful analytic tools for understanding arithmetic se-quences through oscillatory behavior. Classical results by Hua Loo-Keng and R. Vaughan es-tablished fundamental estimates for complete and incomplete Weyl sums, which later became indispensable in deriving asymptotic formulas for higher–degree Diophantine equations and for improving bounds in Waring’s problem. In many applications, it becomes essential to study short Weyl sums, where summation is restricted to a small interval instead of beginning at the origin. These short sums capture more delicate local behavior and appear naturally in problems with “al-most equal” summands, in the analysis of shifted polynomial sequences, and in the investigation of exponential sums on major and minor arcs. Earlier results addressed only small exponents or special cases; however, recent progress has extended these methods to arbitrary fixed degree n. The aim of this article is to refine and simplify the existing analysis of short G. Weyl trigonomet-ric sums on major arcs. Building on the approach of Vaughan and subsequent developments by Rakhmonov and collaborators, we obtain new estimates for the sums T(α; x, y) and establish an asymptotic formula under natural Diophantine conditions. Our results generalize previous work and contribute to a clearer understanding of the structure and behavior of short Weyl sums in analytic number theory.

R. Vaughan [1], studying G. Weyl sums of the form X a 1 T(α, x) = e (αm ) , α = + λ, q ≤ τ, (a, q) = 1, |λ| ≤ , q qτ

used on major arcs the estimate

ak + bk 1 S (a, q) = e ≪ q 2 (b, q), (2.1)

due to Hua Loo-Keng [2], and by means of van der Corput’s method proved:

S(a, q) 1 1

n +ε n 2

T(α, x) = e (λt ) dt + O q 2 (1 + x |λ|) . S(a, q) = S (a, q), Under the condition that α is very well approximated by a rational number with denominator q, that is, when

|λ| ≤ , 2nqx

he also proved:

x S(a, q) 1

n +ε

T(α, x) = e (λt ) dt + O q 2 .

He used these estimates to derive an asymptotic formula in Waring’s problem for eight cubes [3].

Short G. Weyl trigonometric sums of the form

X a 1 T(α; x, y) = e(αm ), α = + λ, q ≤ τ, (a, q) = 1, |λ| ≤ , (2.2) q qτ

x−y<m≤x

obtained from T(α, x) by replacing the condition m ≤ x with x − y < m ≤ x, were investigated on major arcs for n = 2, 3, 4 in [4, 5] and applied in deriving asymptotic formulas with almost equal summands in Waring’s problem (for cubes and fourth powers) and in Estermann’s cubic problem in [6]. Later, for arbitrary fixed n the sum T(α; x, y) was studied in [7, 8]. The main result of this work is a simplification of the proof and refinement of the main theorem from [7, 8].

3 Main part. Statement of the scientific problem.

n−2 n−1 1

Theorem 3.1. Let τ ≥ 2n(n − 1)x y and λ ≥ 0. Then if {nλx } ≤ , the formula

S(a, q) 1 T(α, x, y) = T(λ; x, y) + O(q 2 )

holds, and if {nλx } > , the estimate

|T(α, x, y)| ≪ q n ln q + min (yq n , λ k x k q n )

holds.

Corollary 3.2. Let τ ≥ 2n(n − 1)x y, |λ| ≤ n−1 . Then the relation

2nqx

T(α, x, y) = S(a, q)γ(λ; x, y) + O(q 2 ), Z 0.5 γ(λ; x, y) = e λ x − + yt dt.

−0.5 2

n−2 1 1

Corollary 3.3. Let τ ≥ 2n(n − 1)x y, n−1 < |λ| ≤ . Then we have the estimate

2nqx qτ

1 1 1 1 1

1− − 1− −

T(α, x, y) ≪ q n ln q + min yq n , x k q k n .

Corollaries 3.2 and 3.3 generalize R. Vaughan’s results [1] for short G. Weyl trigonometric sums T(α; x, y) of the form (2.2).

The proof of Theorem 3.1 is carried out by estimating special van der Corput trigonometric sums, using Poisson summation, estimates of trigonometric integrals via the size of derivatives, and the estimate of complete rational sums (2.1) due to Hua Loo-Keng [2].

Proof of Theorem 3.1. Using the orthogonality property of complete linear rational trigono-metric sums, we find

X ak X T(α; x, y) = e + λm 1 =

x−y<m≤x k=1

k≡m(modq)

X ak X = e e(λm ) =

k=1 x−y<m≤x

m≡k(modq)

q n q

X ak X 1 X b(k − m) = e e(λm ) e = q q q

k=1 x−y<m≤x b=1

= Tb(λ; x, y)Sb(a, q), (3.1)

where X bm Tb(λ; x, y) = e λm − , T(λ; x, y) = T0(λ; x, y),

x−y<m≤x

X ak + bk Sb(a, q) = e , S(a, q) = S0(a, q).

Let R(α; x, y) denote the part of T(α; x, y) defined by (3.1), with the term b = q omitted:

R(α; x, y) = Tb(λ; x, y)Sb(a, q). (3.2)

n−1 n−1

Noting that nλx − {nλx } is an integer, represent

Tb(λ; x, y) = e (f(m, b)) ,

x−y<m≤x

n n−1 n−1 bu f(u, b) = λu − (nλx − {nλx })u − .

We compute the first and second derivatives:

′ n−1 n−1 n−1 b f (u, b) = nλ(u − x ) + {nλx } − ,

′′ n−2

f (u, b) = n(n − 1)λu ≥ 0.

Hence f (u, b) is nondecreasing on u ∈ (x − y, x], therefore

′ ′ ′

f (x − y, b) < f (u, b) ≤ f (x, b). (3.3)

Estimating f (x, b) from above:

′ n−1 b b f (x, b) = {nλx } − < 1 − , (3.4)

To obtain a lower bound for f (x − y, b), we use the representation

′ n−1 n−1 n−1 b f (x − y, b) = −nλ x − (x − y) + {nλx } − =

k k n−1−k k n−1

= nλ (−1) Cn−1x y + {nλx } − =

n−2 k k n−1−k k n−1

= −n(n − 1)λx y + nλ (−1) Cn−1x y + {nλx } − . Using the monotonicity of f (u, b), the condition

τ ≥ 2n(n − 1)x y,

and the inequality

k k n−1−k k

W = (−1) Cn−1x y ≥ 0, n ≥ 3, 3x ≥ (n − 3)y,

′ ′ n−1 b f (u, b) ≤ f (x, b) = {nλx } − < 1,

′ ′ n−2 n−1 b f (u, b) ≥ f (x − y, b) = −n(n − 1)λx y + nλW + {nλx } − ≥

n−2 b n(n − 1)x y b 1 ≥ −n(n − 1)λx y − ≥ − − ≥ −1 + .

q qτ q 2q

Therefore, applying the Poisson summation formula to the sum Tb(λ; x, y) with α = −1, β = 1, ε = 0.5, we obtain

Tb(λ; x, y) = I(−1, b) + I(0, b) + I(1, b) + O(1), (3.5)

I(h, b) = e(fh(u, b)) du, fh(u, b) = f(u, b) − hu,

The function ′ n−1 n−1 n−1 b fh(u, b) = nλ(u − x ) + {nλx } − − h

is nondecreasing on the interval u ∈ [x − y, x]. Therefore,

′ ′ ′

fh(x − y, b) ≤ fh(u, b) ≤ fh(x, b),

which may be written in the form

n−1 b ′ n−1 b {nλx } − − h − η < fh(u, b) ≤ {nλx } − − h, (3.6)

n−2 n−2 n(n − 1)x y 1 η = n(n − 1)λx y − nλW ≤ n(n − 1)λx y ≤ ≤ .

qτ 2q

Substituting (3.5) into (3.1) and (3.2), we obtain

T(α; x, y) = T−1 + T0 + T1 + O |Sb(a, q)| , (3.7)

R(α; x, y) = R−1 + R0 + R1 + O |Sb(a, q)| , (3.8)

Th = I(h, b) Sb(a, q),

Rh = I(h, b) Sb(a, q).

Using estimate (2.1), we bound the remainder term:

X X X X 1 1 1 1

− +ε − +ε +ε

|S (a, q)| ≪ q 2 (b, q) = q 2 δ 1 ≤ q 2 τ(q).

b=1 b=1 δ|q 1≤b≤q−1 (b,q)=δ We now estimate each of the sums Th and Rh separately.

Estimate of T1 and R1. Setting h = 1 in (3.6), we obtain

′ n−1 b b f1(u, b) ≤ {nλx } − − 1 ≤ − < 0.

Estimating the integral via the magnitude of the first derivative, we obtain

|I(1, b)| = e(f1(u, b)) du ≪ .

x−y b

From this and from (2.1), we obtain

q−1 q−1 q−1

X X X 1 |Sb(a, q)| 1 (b, q) 1

+ε +2ε

R = I(1, b)S (a, q) ≪ ≪ q 2 ≪ q 2 .

q b b

b=1 b=1 b=1

In the case b = 0, using the inequality

(k) n−k n−k

f (u, q) ≥ n(n − 1) · · · (n − k + 1)λ(x − y) ≫ λx , k = 2, 3, . . . , n,

and estimating the integral I(1, 0) via the magnitude of the kth derivative, we find

|I(1, 0)| ≪ min y, λ k x k .

From this, and using the estimate |S(a, q)| ≪ q n , taking into account the estimate for R ,

|I(1, 0)| |S(a, q)| 1 1 1 n 1

+2ε − − 1− −

T ≤ |R | + ≪ q 2 + min yq n , λ k x k q n .

q 2≤k≤n

Estimate of T−1 and R−1. Setting h = −1 in (3.6), we obtain

′ n−1 q − b q − b f−1(u, b) > {nλx } + − η ≥ .

The integral I(−1, b) is also estimated using the magnitude of the first derivative. We have

|I(−1, b)| = e(f−1(u, b)) du ≪ .

x−y q − b

Proceeding analogously to the case of estimating R1, we obtain

q−1 q−1 q−1

X X X I(−1, b) Sb(a, q) |Sb(a, q)| 1 (b, q) 1

+ε +2ε

R = ≪ ≪ q 2 ≪ q 2 .

q q − b b

b=1 b=1 b=1

|I(−1, 0)| |S(a, q)| 1 |S(a, q)| 1

+2ε +2ε

T ≤ |R | + ≪ q 2 + ≪ q 2 .

−1 −1

Estimate of R0. If {nλx } ≤ , then setting h = 0 in (3.6), we obtain

′ n−1 b 1 − 2b b f0(u, b) ≤ {nλx } − ≤ ≤ − < 0.

q 2q 2q

Estimating the integral I(0, b) again using the magnitude of the first derivative, we find

|I(0, b)| = e(f0(u, b)) du ≪ .

x−y b

Proceeding similarly to the estimate for R1, we obtain

q−1 q−1 q−1

X X X I(0, b) Sb(a, q) |Sb(a, q)| 1 (b, q) 1

+ε +2ε

R = ≪ ≪ q 2 ≪ q 2 .

q b b

b=1 b=1 b=1 From this, together with the estimates for R1 and R−1 and using (3.8), we obtain the first assertion of the theorem.

Estimate of T0. When {nλx } ≥ , we define a natural number r by the relation

r n−1 r + 1 ≤ {nλx } < , 1 ≤ r ≤ 2q − 1.

2q 2q

From this, using inequality (3.6) with h = 0 and the condition η ≤ , we obtain

′ n−1 b r − 2b − 1 f0(u, b) > {nλx } − − η ≥ , (3.9) q 2q

′ n−1 b r − 2b + 1 f0(u, b) ≤ {nλx } − < . (3.10) q 2q

Let r = 2r1 be even (1 ≤ r1 ≤ q − 1). We split the summation interval 0 ≤ b ≤ q − 1 in the sum T0 into the following three sets:

0 ≤ b ≤ r1 − 1, b = r1, r1 + 1 ≤ b ≤ q − 1,

in the first of which the right-hand side of inequality (3.9) is positive, and in the third the right-hand side of (3.10) is negative, that is,

′ 2r1 − 2b − 1 r1 − b f0(u, b) > ≥ , 0 ≤ b ≤ r1 − 1, 2q 2q

′ 2r1 − 2b + 1 r1 − b f0(u, b) < ≤ , r1 + 1 ≤ b ≤ q − 1.

2q 2q

Using these inequalities and estimating the integral I(0, b) via the size of the first derivative,

I(0, b) = e(f0(u, b)) du ≪ , b ̸= r1.

x−y |r1 − b|

In the case b = r1, evaluating as in the estimate of the integral I(1, 0), we find

|I(0, r )| ≪ min y, λ k x k .

Using these estimates together with the bound |S(a, q)| ≪ q n , we obtain

I(0, b)Sb(a, q) 1  q 1 n 

− − 1−

T = ≪ q n + min y, λ k x k 0   q |r1 − b| 2≤k≤n

b=0 b=0 b̸=r1

≪ q n ln q + min yq n , λ k x k q n .

Let now r = 2r1 + 1 be odd (0 ≤ r1 ≤ q − 1). We split the summation interval 0 ≤ b ≤ q − 1 in the sum R0 into the following three sets:

0 ≤ b ≤ r1 − 1, b = r1, r1 + 1, r1 + 2 ≤ b ≤ q − 1,

in the first of which the right-hand side of inequality (3.9) is positive, and in the third the right-hand side of (3.10) is negative, that is,

′ 2r1 + 1 − 2b − 1 r1 − b f0(u, b) > = , 0 ≤ b ≤ r1 − 1, 2q q

′ 2r1 + 1 − 2b + 1 r1 − b f0(u, b) < ≤ , r1 + 2 ≤ b ≤ q − 1.

2q 2q Consequently,

I(0, b) = e(f0(u, b)) du ≪ , b ̸= r1 − 1, r1.

x−y |r1 − b|

In the case b = r1 − 1, r1, proceeding as in the previous estimate of I(0, r1), we obtain

|I(0, b)| ≪ min y, λ k x k , b = r , r + 1.

From these estimates for I(0, b) we obtain

T0 ≤ |I(0, b)| |Sb(a, q)|

1  q 1 n 

− − 1−

≪ q n + min y, λ k x k |r1 − b| 2≤k≤n

b̸=r1, r1+1

≪ q n ln q + min yq n , λ k x k q n .

Substituting the estimates for T1, T−1, and T0 into (3.7), we obtain the second assertion of the theorem.

REMARK. The case λ < 0 reduces to the case λ ≥ 0 if we rewrite formula (3.1) in the form

1 X 1 X T(α; x, y) = Tq−b(−λ; x, y) Sq−b(q − a, q) = Tb(−λ; x, y) Sb(q − a, q).

b=0 b=0

In this work, we investigated the behavior of short G. Weyl trigonometric sums on major arcs and refined earlier results obtained by Vaughan and subsequent researchers. By applying a com-bination of analytical tools-including Poisson summation, derivative-based estimates for oscilla-tory integrals, and Hua Loo-Keng’s classical bounds for complete rational sums—we achieved a clearer and more streamlined derivation of the principal estimates governing these short sums. The approach emphasizes the advantages of analyzing the structure of derivatives and exploiting the monotonicity properties inherent in the phase functions associated with the sums. The main theorem provides two distinct outcomes depending on the approximation properties of the fre-quency parameter. In the region where the approximation is sufficiently strong, a precise relation between the short Weyl sum and its continuous analogue emerges, allowing the sum to be effec-tively expressed in terms of complete rational sums. In the complementary region, we obtained nontrivial bounds that extend and generalize Vaughan’s earlier estimates. These results not only deepen the understanding of the analytic behavior of short Weyl sums but also strengthen their applicability to classical additive problems, particularly those related to Waring-type questions. Overall, the methods developed here demonstrate that a careful combination of classical analytic number theory techniques can yield significant simplifications and improvements to previously established results. This contributes to a broader framework for approaching multidimensional exponential sums and enhances the toolkit available for tackling related problems in modern analytic number theory.

The authors express their sincere gratitude to the reviewers and editors for their useful comments.