Fourier Sparsity in Delta Functions

• Fourier sparsity of delta functions is defined as the number of nonzero Fourier coefficients for functions on the Boolean hypercube.
• The established bounds, including a lower bound of r+1 and exponential upper bounds, have significant implications for Matching Vector PIR schemes.
• Techniques such as support-sum and polynomial identity arguments bridge algebraic and Fourier-analytic methods to derive these sparsity results.

A delta function on the Boolean hypercube $\{0,1\}^r \subseteq \mathbb{Z}_m^r$ is defined as a function $f: \mathbb{Z}_m^r \to \mathbb{C}$ with $f(0) = 1$ and $f(x) = 0$ for every nonzero $x \in \{0,1\}^r$. The Fourier sparsity of such a delta function refers to the cardinality of its support in the group-theoretic Fourier basis, i.e., the number of nonzero Fourier coefficients. Questions concerning the minimum achievable Fourier sparsity for such delta functions are of foundational interest and have critical implications for algebraic-combinatorial protocols such as Matching Vector-based Private Information Retrieval (PIR) schemes (Ghasemi et al., 5 Dec 2025).

1. Formal Framework and Definitions

Let $m \ge 2$ and $r \ge 1$. Define $G = \mathbb{Z}_m^r$, and the Boolean hypercube as $$\{0,1\}^r = \{(x_1, \ldots, x_r) \in \mathbb{Z}_m^r : x_i \in \{0,1\}\}$$. A delta function on this hypercube is $f : G \to \mathbb{C}$ with

$$f(0) = 1, \qquad f(x) = 0 \text{ for every nonzero } x \in \{0,1\}^r.$$

The Fourier basis of $G$ consists of the characters

$$\chi_\alpha(x_1, \dots, x_r) = \prod_{i=1}^r \omega^{\alpha_ix_i},$$

where $\omega = e^{2\pi i/m}$ is a primitive $m$th root of unity and $$\alpha = (\alpha_1, ..., \alpha_r) \in \mathbb{Z}_m^r$$. The (normalized) Fourier coefficients are

$$\widehat f(\alpha) = \frac{1}{|G|} \sum_{x \in G} f(x) \overline{\chi_\alpha(x)},$$

with the expansion $$f(x) = \sum_{\alpha \in G} \widehat f(\alpha)\chi_\alpha(x)$$. The Fourier support is $$\operatorname{Supp}(\widehat f) = \{\alpha \in G : \widehat f(\alpha) \neq 0\}$$, and the sparsity is defined as $$\|\widehat f\|_0 = |\operatorname{Supp}(\widehat f)|$$ (Ghasemi et al., 5 Dec 2025).

2. Sparsity Bounds: Main Results

For $f$ a delta function on $\{0,1\}^r \subset \mathbb{Z}_m^r$:

• General Lower Bounds:

$\|\widehat f\|_0 \ge r + 1$

$\|\widehat f\|_0 \ge \left(\frac{m}{m-1}\right)^r$

• Explicit Upper Bounds:
  • If $m > r$:

  $\exists\, f: \|\widehat f\|_0 = r+1$ - If $(m-1) \mid r$:

  $\exists\, f: \|\widehat f\|_0 \leq m^{\,r/(m-1)}$

For fixed $m$, as $r \to \infty$: $$\left(\frac{m}{m-1}\right)^r \lesssim \|\widehat f\|_0 \lesssim \exp\left(\frac{\ln m}{m-1}\,r\right)$$

Specialized to $m=3$: $\Omega(1.5^r) \le \|\widehat f\|_0 \le O(\sqrt{3}\,^r)$

These bounds constrain how "concise" a delta function can be in the Fourier basis. The precise base of exponential sparsity is not generally determined, particularly for small $m$ (Ghasemi et al., 5 Dec 2025).

3. Techniques for Bounds: Proof Strategies and Constructions

• Lower Bounds ("Support-Sum Argument"):

Construct auxiliary

$$g(x) = \prod_{i=1}^r \left(\omega^{x_i} - \omega^2\right) \cdot \prod_{i=1}^r\prod_{j=2}^{m-1}(\omega^{x_i} - \omega^j)$$

supported on $\{0,1\}^r$, $g(0) \neq 0$, and with Fourier support $\{1, ..., m-1\}^r$. Multiplying $f \cdot g$ yields the total delta on $G$, whose Fourier support is all of $G$. The convolution/support-sum lemma implies:

$$\operatorname{Supp}(\hat f) + \{1, ..., m-1\}^r = G$$

Forcing $\|\widehat f\|_0 \geq r+1$ via a diagonal argument; a volume argument gives $\|\widehat f\|_0 \ge (m/(m-1))^r$.

• Lower Bounds ("Polynomial Identity Argument"):

Write $$f(x) = \sum_{j=1}^t b_j \prod_{i=1}^r \alpha_i(j)^{x_i}$$ and analyze the vanishing constraints on $\{0,1\}^r \setminus \{0\}$, leading to formal identities that cannot be satisfied for $t \le r$.

• Upper Bounds (Explicit Constructions):
  • For $m > r$, set

  $$f(x_1, ..., x_r) = \prod_{i=1}^r (\omega^{x_1 + \cdots + x_r} - \omega^i)$$

  which vanishes as required on $\{0,1\}^r \setminus \{0\}$ and is supported on $r+1$ characters $(i, ..., i)$ for $i = 0, 1, ..., r$. - For $(m-1) \mid r$, partition the coordinates into $\ell = r/(m-1)$ blocks of size $m-1$, constructing block functions with $m$-sparse Fourier support, multiply across blocks, yielding an overall support of at most $m^\ell$ (Ghasemi et al., 5 Dec 2025).

4. Relevance to Matching Vector PIRs and S-Decoding Polynomials

In Matching Vector PIR schemes, the main ingredients are:

• An $S$-matching vector family over $\mathbb{Z}_m$,
• An $S$-decoding polynomial $P \in F[Z]$ vanishing on $\{\gamma_m^s : s \in S \setminus \{0\}\}$ with $P(1) = 1$.

A $t$-sparse $S$-decoding polynomial produces a $t$-server PIR with communication $O(k)$. In known constructions, $$S = \{s \in \mathbb{Z}_m : s^2 \equiv s \bmod m\} \cong \{0,1\}^r$$ (for $m = p_1 \cdots p_r$).

A $t$-sparse $S$-decoding polynomial induces a delta function on $$\{0,1\}^r \subseteq \mathbb{Z}_{p_1} \times \cdots \times \mathbb{Z}_{p_r}$$ of Fourier sparsity $t$. Hence, the bounds on delta function sparsity show that any such PIR must use at least $r+1$ servers, or

$t \geq r+1,$

with communication complexity lower bounded by $\exp((\log n)^{1/(r+1)})$. This rules out, via improved $S$-decoding polynomials, schemes with polylogarithmic communication and constant servers for the known matching vector families (Ghasemi et al., 5 Dec 2025).

5. Contrasts and Related Notions: Dirac Combs and Fractional Fourier Transforms

The phenomenon of Fourier sparsity for delta functions extends to the setting of Dirac combs and fractional Fourier analysis in the continuous domain. For Dirac combs

$$_r(x) = \sqrt r\sum_{k\in\mathbb{Z}}\delta(x - rk),$$

the fractional Fourier transform $\mathcal{F}^\alpha$ preserves discrete support if and only if $r\,\cos(\frac{\pi\alpha}{2})$ and $(1/r)\,\sin(\frac{\pi\alpha}{2})$ are $\mathbb{Z}$-linearly dependent. In such cases, $\mathcal{F}^\alpha[_r]$ becomes a sparse sum of delta functions with coefficients governed by theta functions and explicit Gauss sums. This underlines a deep connection between algebraic criteria for sparsity and the analytic structure of the fractional Fourier transform, with the metaplectic representation and theta function identities playing critical roles (Viola, 2018).

6. Open Problems and Research Directions

Several major questions remain unresolved:

• For $m = p_1\cdots p_r$ with distinct primes, does there exist a delta function with precisely $r+1$ Fourier support? Equivalently, can one construct $(r+1)$-sparse $S$-decoding polynomials in this setting? A positive answer would improve Matching-Vector PIR communication to $\exp((\log n)^{1/(r+1)})$.
• In the complex-valued case with distinct primes, it is conjectured that the sparsity cannot be improved beyond the trivial $2^r$, proved for $r=2$ via a Möbius/S-lemma. The general case remains open and is believed to be more difficult.
• For uniform $\mathbb{Z}_m^r$ with small $m$, the exact exponential base for the minimal sparsity is known to lie between $\sqrt[m-1]{m}$ and $m/(m-1)$, but the sharp value remains undetermined.

Resolving these would clarify the potential for future algebraic-combinatorial design of PIR and locally decodable codes, sharpening the interface between Fourier-analytic and combinatorial structures (Ghasemi et al., 5 Dec 2025).