arXiv:2608.03194v1

Strichartz estimates for quasi-periodic functions on long time intervals: An arithmetic approach

Kotaro Inami

math.APmath.CAmath.NT42B3742A7511J68

Paper metadata

Original submission: August 4, 2026
Claimed MSC: 42B37, 42A75, 11J68
arXiv categories: math.AP, math.CA, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 25 pages

Abstract

We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell--Vinogradov systems.

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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 15, 2026

01Statements4 reported findingsCorrect

The long-time Strichartz estimates for quadratic and higher-degree algebraic frequency ratios, their torus reformulation, and the endpoint fractional estimate are correct. The degree-dependent time scales follow from the verified lattice-energy counts, and the endpoint loss follows from the cited reverse square-function theorem together with the height and frequency decompositions.

Theorem 2Correct

Quadratic-frequency long-time estimate

Page 4 · Theorem 2, Equation (5) · arXiv:2608.03194v1

For an algebraic irrational $\omega$ of degree $2$ , the sixth-moment expansion reduces to counting triples in $\mathbb Z+\omega\mathbb Z$ with fixed sum and nearly fixed sum of squares. Lemma 6 gives $C^{\varepsilon}$ choices at each exact energy, while Roth's theorem gives $C^{\varepsilon}$ admissible energies in a window of width $C^{-2}$ . The remaining dyadic energies contribute the same $C^{2+\varepsilon}$ sixth-power bound after time integration. Thus the estimate on an interval of length $C^2$ has factor $C^{1/3+\varepsilon}$ ; interpolation with the height-Bernstein bound and subdivision of $[0,T]$ give exactly $\left(1+\left(\frac{T}{C^2}\right)^{1/p}\right)C^{1-4/p+\varepsilon}.$

Full paper, version 1 ↗

Theorem 3Correct

Higher-degree long-time estimates

Page 5 · Theorem 3, Equations (6)–(7) · arXiv:2608.03194v1

When $\deg\omega>2$ , the numbers $1,\omega,\omega^2$ are linearly independent over $\mathbb Q$ . The multivariate Roth bound therefore gives $C^{2+\varepsilon}$ possible near-resonant energies at scale $C^{-2}$ and only $C^{\varepsilon}$ at scale $C^{-4}$ . These are precisely the losses required for the two regimes in Equation (6). At the fourteenth moment, the Bourgain–Demeter quadratic Parsell–Vinogradov estimate and the same Diophantine count yield $C^{\varepsilon}\left(C^{5/7}+T^{1/14}C^{3/7}\right),$ and interpolation with the supremum norm gives Equation (7) for every $p\geq14$ .

Bourgain–Demeter, quadratic Parsell–Vinogradov estimate ↗

Theorem 6Correct

Endpoint fractional Strichartz estimate

Page 6 · Theorem 6, Equation (8) · arXiv:2608.03194v1

At frequency scale $N$ , the cited reverse square-function theorem partitions the curve at horizontal scale $N^{-1}$ when $a\geq2$ and $N^{-a/2}$ when $0<a<2$ , $a\neq1$ . A corresponding frequency slab contains at most $O_{\vec\omega}(C^{\nu-1})$ lattice points in the first case and $O_{\vec\omega}(N^{1-a/2}C^{\nu-1}+1)$ in the second. The $L^2$ -to- $L^4$ Bernstein estimate therefore gives the factor $C^{(\nu-1)/4}\left(1+N^{\sigma_a}\right),\qquad \sigma_a=\begin{cases}0,&a\geq2,\\(2-a)/8,&0<a<2,\ a\neq1.\end{cases}$ Frequency and height square-function summation then produces the stated Sobolev exponent $(\nu-1)/4+\sigma_a$ , including the endpoint.

Bulj–Inami–Shiraki, Theorem 1.2 ↗

Corollary 1Correct

Degenerate torus reformulation

Pages 5–6 · Corollary 1 · arXiv:2608.03194v1

Lemma 1 identifies the spatial mean norm of a two-frequency quasi-periodic polynomial with the corresponding norm on $\mathbb T^2$ . Under this identification, the phase $(k_1+\omega k_2)^2$ is exactly the symbol of the displayed rank-one operator $\widetilde\Delta$ . Substitution of Theorems 2 and 3 gives all three clauses with the printed powers of $T$ and $C$ .

02Proofs8 reported findingsCorrect

The central proof chains are correct and complete after the uniquely determined typographical repairs listed below. The exact-energy lemma reduces to divisor and oval lattice-point counts; the higher-degree argument uses the stated Schmidt-subspace estimate with the correct separation exponents; the fourteenth-moment input follows from Bourgain–Demeter decoupling; and the endpoint argument applies the reverse square-function theorem at its supported scales.

Lemma 6 and proof of Theorem 2Correct and complete

Exact and near-energy counting

Pages 12–16 · Lemma 6 and proof of Theorem 2 · arXiv:2608.03194v1

For $\deg\omega>2$ , comparison of the coefficients of $1,\omega,\omega^2$ reduces the exact fiber to two positive-definite Eisenstein norm equations, each having $C^{\varepsilon}$ representations. In the quadratic case, the corrected identities recorded in the typo finding below put $(Q(N),\Delta(K,N))$ on a fixed dilate of an ellipse; the cited oval estimate gives $C^{\varepsilon}$ possibilities, the norm equation gives $C^{\varepsilon}$ choices of $N$ , and the bilinear equation together with $\Delta$ determines $K$ . Roth separation then bounds the number of nearby energy values. The same separation applied to differences of two admissible coefficient pairs supplies the elementary packing step compressed on pages 14 and 16.

Bombieri–Pila, integral points on analytic ovals ↗

Proof of Theorem 3 for $6\leq p<14$Correct and complete

Two Diophantine time scales

Pages 17–19 · Section 5.1 · arXiv:2608.03194v1

Writing an energy as $t_1+\omega t_2+\omega^2t_3$ , with $|t_i|\lesssim C^2$ , the multivariate Roth inequality makes distinct coefficient triples in a window of width $C^{-2}$ separated by $C^{2/(2+\varepsilon)}$ and those in a window of width $C^{-4}$ separated by $C^{4/(2+\varepsilon)}$ . Packing in the two unconstrained coefficient directions gives respectively $C^{2+\varepsilon}$ and $C^{\varepsilon}$ values. Lemma 6 controls every exact fiber, and the dyadic nonresonant sums have the required $AC^{4+\varepsilon}$ bound. Taking sixth roots, interpolating, and subdividing time yields both regimes of Equation (6).

Proof of Theorem 3 for $p\geq14$Correct and complete

Parsell–Vinogradov and restricted-input reduction

Pages 19–21 · Section 5.2 · arXiv:2608.03194v1

The paper's inhomogeneous subset estimate $J_s(N,S)\lesssim_{\varepsilon,s}N^{\varepsilon}\left(N^{2s-2}+N^{4s-10}\right)|S|$ follows from the cited Bourgain–Demeter discrete restriction estimate: for $s\leq4$ its moment bound is $N^{\varepsilon}|S|^s\leq N^{2s-2+\varepsilon}|S|$ , while for $s\geq4$ it is $N^{2s-8+\varepsilon}|S|^s\leq N^{4s-10+\varepsilon}|S|$ . A shifted right-hand side is a Fourier coefficient of the same nonnegative moment and is bounded by the zero coefficient. After six of the fourteen variables are fixed, the case $s=4$ gives $C^6|S|^7$ . Counting the remaining energy coefficients at each dyadic scale produces $C^{\varepsilon}(C^{10}+TC^6)|S|^7$ , whose fourteenth root is the claimed estimate. Lemma 7 then transfers the restricted estimate to arbitrary coefficients.

Bourgain–Demeter, Theorems 1.1–2.2 and Corollary 2.3 ↗

Proposition 2 and proof of Theorem 6Correct and complete

Reverse square-function application and dyadic summation

Pages 21–24 · Proposition 2 and proof of Theorem 6 · arXiv:2608.03194v1

After the rescaling $(x,t)\mapsto(N^{-1}x,N^{-a}t)$ , the thickening of the curve has width comparable to $N^{-a}$ . The cited theorem applies with partition exponent $b=a$ for $a\geq2$ and $b=2$ for $0<a<2$ , so its loss exponent is zero in both cases. Finite-overlap enlargement of the projected slabs, the vector-valued Hilbert-transform bound, and the weighted spatial-mean identity reduce the estimate to lattice points in a single slab. The elementary choice of one coordinate with nonzero $\omega_j$ verifies the stated slab counts. Plancherel and the two Littlewood–Paley summations then give exactly Equation (8). Frequencies $N<1$ , for which the displayed rescaling is unnecessary, follow directly from the same $L^2$ -to- $L^4$ slab bound and introduce no additional loss.

Bulj–Inami–Shiraki, reverse square-function estimate ↗

Lemma 6, quadratic casesTypo

Two scale factors in the conic identities are extraneous

Pages 12–13 · proof of Lemma 6, Cases 2-1 and 2-2 · arXiv:2608.03194v1

In Case 2-1 the displayed identity has an extra factor $a$ before the first square. From the two preceding equations and $4Q(K)Q(N)-L_2^2/4=3\Delta(K,N)^2$ , the correct identity is $\left(4cQ(N)+L_1\right)^2-12ac\Delta(K,N)^2=L_1^2+acL_2^2.$ In Case 2-2, with $D=b^2-4ac$ and $A=2aL_1-bL_2$ , the right-hand side must be $A^2-DL_2^2$ , not $DA^2-DL_2^2$ . Direct expansion uniquely determines both corrections. Each corrected equation is still an ellipse equation, so the lattice-point argument is unchanged.

Proof of Theorem 2Typo

The coefficient norm is omitted from two intermediate target displays

Pages 14 and 16 · Step 1 after the near-resonant integral and Step 2 before Equation (21) · arXiv:2608.03194v1

The first “it is enough to show” display ends with $C^{\varepsilon}$ , and the later display ends with $AC^{2+\varepsilon}$ , without the homogeneous factor. In both places the right-hand side must also contain $\left(\sum_{\lambda}|a_{\lambda}|^2\right)^3.$ The immediately following Cauchy–Schwarz displays contain exactly this factor and prove the corrected inequalities, so the omission is local and harmless.

Proof of the fourteenth-moment estimateTypo

One occurrence of the quadratic energy drops the exponent

Page 21 · display immediately after the dyadic energy decomposition · arXiv:2608.03194v1

The counting display prints $|\alpha+2\beta\omega+\gamma\omega|\sim2^j$ . It must read $|\alpha+2\beta\omega+\gamma\omega^2|\sim2^j$ , as in the preceding and following displays and as dictated by $(k_1+\omega k_2)^2$ . This unique correction restores the linear form to which the multivariate Roth estimate is applied.

Proposition 2, Equation (28)Typo

The time cutoff and time domain are mismatched in one square-function display

Page 23 · Minkowski display and Equation (28); repeated in the second case on page 24 · arXiv:2608.03194v1

The line obtained from Minkowski drops the factor $\eta(t)$ even though it occurs immediately before and after that line; it must be retained. Correspondingly, the right-hand norm in Equation (28) is printed over $[0,1]\times\mathbb R$ although it contains $\eta(t)$ and the adjacent formulas use $\mathbb R\times\mathbb R$ . Its time domain must be $\mathbb R$ . With these mechanically determined corrections, the subsequent Bernstein estimate is on the same domain and the original $[0,1]$ estimate follows because $|\eta|$ is bounded below there.

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Paper: arXiv:2608.03194v1
Authors listed: Kotaro Inami
Audit date: August 15, 2026

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P(delta)

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Strichartz estimates for quasi-periodic functions on long time intervals: An arithmetic approach

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Quadratic-frequency long-time estimate

Higher-degree long-time estimates

Endpoint fractional Strichartz estimate

Degenerate torus reformulation

Exact and near-energy counting

Two Diophantine time scales

Parsell–Vinogradov and restricted-input reduction

Reverse square-function application and dyadic summation

Two scale factors in the conic identities are extraneous

The coefficient norm is omitted from two intermediate target displays

One occurrence of the quadratic energy drops the exponent

The time cutoff and time domain are mismatched in one square-function display

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