Common Spectral Eigenvalue Spectra for Random Convolutions Generated by Hadamard Triples

Abstract: Lu proved that the set $$\mathcal{T}:={t\in\mathbb{Z}\setminus{0}:(q,\mathcal{D},tL)\text{ forms a Hadamard triple}}$$ constitutes a spectral eigenvalue set for $μ<em>{q,D}$, where $(q,\mathcal{D},L)$ is a Hadamard triple. And they prove for $s \in [0,\frac{\log #\mathcal{D}}{\log q}]$, the corresponding spectra form a family of cardinality continuum. In this paper, we study Moran measures formed by random convolutions of finite Hadamard triples. Let $$μ=δ</em>{M_1^{{-1}D_1}<em>δ_{M_2^{-1}D_2}</em>\cdots,} \qquad M_k=q_1q_2\cdots q_k,$$ where the factors are produced from finitely many triples ${(N_j,B_j,L_j):1\le j\le m}$, $$(ω<em>k)</em>{k=1}^{{\infty}\in{1,2,\ldots,m}^{\mathbb} N}$$, and $n_k\in\mathbb N<sup>+$, by setting $$q_k=N_{ω<em>k}^{n_k},\qquad D_k=B</em>{ω<em>k},\qquad E_k=N</em>{ω<em>k}^{n_k-1}L</em>{ω<em>k}.$$ Assume a non-full-digit gap $$ρ:=\min</em>{1\le j\le m}\frac{N_j}{#B_j}&gt;1.$$ For the common Hadamard triple multiplier set $$\mathcal{T}<em>*:=\bigcap</em>{j=1}^{m{t\in\mathbb{Z}\setminus{0}:(N_j,B_j,tL_j)\text{} is a Hadamard triple}},$$ Our main result is that, for every $$0\le s\le κ<em>ω:=\limsup</em>{R\to\infty}\frac{\sum_{r=1}^R\log #D_r}{\sum_{r=1}^R\log q_r},$$ there exist continuum many countable sets $Λ\subset\mathbb{Z}$ such that $tΛ$ is a spectrum of $μ$ for every $t\in\mathcal{T}<em>*$ and $\dim</em>{Be}Λ=s$.

• The paper proves that one-dimensional random convolutions generated by finitely many Hadamard triples with gap condition ρ>1 have continuum many spectra of every Beurling dimension s∈[0,κω], simultaneously for all multipliers in the common set T*.
• The proof combines a Hadamard-triple Fourier counting identity, uniform lower bounds for tail Fourier transforms, and an inductive carry-adjusted digit construction with sparse branches to control spectrality and dimension.
• The result removes the earlier coprimality requirement and establishes T* as a spectral eigenvalue set, while leaving the full-digit case ρ=1, higher-dimensional extensions, and stronger geometric properties of the spectra open.

Setting and main result

This paper concerns spectral measures arising from infinite random convolutions of finite Hadamard triples. A Borel probability measure $\mu$ on $\mathbb{R}^d$ is spectral if there exists a countable set $\Lambda$ such that the exponentials $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$ form an orthonormal basis of $L^2(\mu)$. The paper works in one dimension with finitely many Hadamard triples $(N_j,B_j,L_j)$, $1\le j\le m$, where $(N,\mathcal{D},L)$ is a Hadamard triple when the matrix $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$ is unitary. Fixing a symbol sequence $$\omega=(\omega_k)\in\Omega_m=\{1,\dots,m\}^{\mathbb{N}}$$ and block lengths $\mathbb{R}^d$0, the measure under study is the Moran-type convolution

$\mathbb{R}^d$1

Li–Miao–Wang established spectrality of such measures for every $\mathbb{R}^d$2 when $\mathbb{R}^d$3; the present work removes that coprimality hypothesis from the construction by imposing instead a non-full-digit gap condition $\mathbb{R}^d$4. The main theorem states: for every $\mathbb{R}^d$5, where

$\mathbb{R}^d$6

is the upper entropy dimension of $\mathbb{R}^d$7, there exist continuum many countable sets $\mathbb{R}^d$8 such that $\mathbb{R}^d$9 is a spectrum of $\Lambda$0 simultaneously for every $\Lambda$1 in the common multiplier set

$\Lambda$2

and $\Lambda$3. Consequently $\Lambda$4 is a spectral eigenvalue set for $\Lambda$5. This extends Lu's result for self-similar measures (recovered when all triples coincide and $\Lambda$6), and it answers affirmatively, within this class, the interpolation question of whether spectra exist at every Beurling dimension up to the upper entropy dimension — a question sharpened by Shi's upper bound $\Lambda$7 and An–Lai's zero-dimension examples. Note that $\Lambda$8 always contains all nonzero integers coprime to $\Lambda$9, so the conclusion is non-vacuous.

Key analytic ingredients

The proof rests on three technical pillars.

Hadamard triple counting identity. If both $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$0 and $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$1 are Hadamard triples, then $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$2 for all $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$3. Since $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$4, this forces at least $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$5 of the sampled values to be nonzero. Combined with the finite-zero lemma of Li–Miao–Wang (the set of zeros of all $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$6 in a compact interval is finite), the gap condition $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$7 guarantees that along any word $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$8 of length $E(\Lambda)=\{e^{2\pi i\lambda\cdot x}\}$9, at least two integers $L^2(\mu)$0 yield nonzero Fourier samples. A compactness argument over $L^2(\mu)$1, using equicontinuity of $L^2(\mu)$2 and continuity of $L^2(\mu)$3 (proved here via uniform tail estimates), yields uniform constants $L^2(\mu)$4 such that every tail measure has two "good" frequencies near each $L^2(\mu)$5. Crucially, a refinement (Lemma 3.5) shows these good frequencies can be chosen outside any prescribed finite set and beyond any radius $L^2(\mu)$6, via periodicity of the finite-block Fourier transform — this is what permits the sparse-branch construction needed for dimension control.

Inductive spectrum construction. The naive candidate $L^2(\mu)$7 is too rigid; the authors exploit the mod-$L^2(\mu)$8 invariance of Hadamard triples to adjust digits by carries $L^2(\mu)$9 selected so that tail Fourier transforms stay bounded below near scaled points $(N_j,B_j,L_j)$0. Each layer splits into a "principal" part $(N_j,B_j,L_j)$1 (built from a chosen subset $(N_j,B_j,L_j)$2 containing $(N_j,B_j,L_j)$3) and a "sparse" part $(N_j,B_j,L_j)$4 whose elements grow geometrically ($(N_j,B_j,L_j)$5). Orthogonality of $(N_j,B_j,L_j)$6 for $(N_j,B_j,L_j)$7 follows case-by-case from the block Hadamard triple property, and cardinality matching ($(N_j,B_j,L_j)$8) upgrades orthogonality to completeness. The classical spectral criterion of An–He–He then lifts finite-layer spectra to spectra of $(N_j,B_j,L_j)$9: a traversal sequence visiting each $1\le j\le m$0 infinitely often ensures the uniform lower bound on tails holds along a subsequence for each fixed $1\le j\le m$1.

Dimension control. Lemma 4.2 selects the principal blocks $1\le j\le m$2 so that $1\le j\le m$3: trivially for $1\le j\le m$4 or $1\le j\le m$5, and otherwise by choosing $1\le j\le m$6 large enough that $1\le j\le m$7, with rounding errors summable. Since the sparse part has Beurling dimension zero, $1\le j\le m$8 reduces to an upper bound on counting functions of $1\le j\le m$9, obtained through a digit-expansion argument showing at most two admissible tail sequences per window and controlling carry choices by $(N,\mathcal{D},L)$0, which is negligible because levels are taken sufficiently far apart.

Continuum many spectra

The multiplicity result uses binary branching at infinitely many marker stages: for each binary sequence $(N,\mathcal{D},L)$1, one of the two good frequencies supplied by Corollary 3.3 / Lemma 3.5 is chosen as a marker, while the skipped one enters a permanent forbidden list. Distinct sequences produce distinct spectra, since a marker chosen in branch $(N,\mathcal{D},L)$2 can never reappear in branch $(N,\mathcal{D},L)$3 — dense branches only extend previously chosen points, and sparse branches respect the forbidden lists. As all constructed sets are subsets of $(N,\mathcal{D},L)$4, the family has cardinality exactly $(N,\mathcal{D},L)$5.

Limitations and open questions

The main theorem requires the strict gap $(N,\mathcal{D},L)$6; the behavior at $(N,\mathcal{D},L)$7 (full-digit sets, e.g. $(N,\mathcal{D},L)$8) is not covered, and the example given ($(N,\mathcal{D},L)$9, $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$0, $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$1) shows the gap condition is genuinely weaker than the coprimality assumption $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$2 of prior work but does not address its necessity. All results are one-dimensional; no higher-dimensional analogue is attempted. The construction also presumes the spectrality framework of Li–Miao–Wang, so measures outside the Hadamard-triple-generated class remain untouched. Two natural questions left open are whether the simultaneous-spectra conclusion persists without the gap condition, and whether the continuum-many statement can be strengthened to spectra with additional structure (e.g., uniformly discrete or relatively dense).

Conclusion

The paper proves that for random convolutions generated by finitely many Hadamard triples satisfying a non-full-digit gap, the common multiplier set $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$3 is a spectral eigenvalue set admitting continuum many spectra at every Beurling dimension between $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$4 and the upper entropy dimension $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$5. The argument combines a Fourier-sampling identity for Hadamard triples, compactness-based uniform lower bounds on tail transforms, an inductive carry-adjusted lattice construction, and a binary-branching scheme for cardinality. It thereby extends the self-similar theory of Lu to the Moran/random setting and removes the coprimality restriction of earlier spectrality results, at the cost of the gap assumption $$\frac{1}{\sqrt{\#\mathcal{D}}}(e^{-2\pi i d\ell/N})_{d\in\mathcal{D},\ell\in L}$$6.