Tuxanidy–Panario Equidistribution Estimate

• The paper establishes that base‑b palindromes are nearly uniformly distributed modulo d² for all square‑free d up to x^(1/4–ε) with a strong logarithmic error saving.
• It employs Fourier expansion and a Baier–Zhao large sieve for square moduli to convert L² bounds into uniform (L∞) error estimates.
• This result underpins analytic number theory advances in counting square‑free palindromes and inspires further research on sparse digital structures.

The Tuxanidy–Panario equidistribution estimate is a quantitative result on the uniformity of base-$b$ palindromes modulo small square moduli. Its principal application is in analytic number theory, particularly in counting palindromic numbers with additional arithmetic constraints, such as square-freeness. The core assertion is that, subject to mild coprimality restrictions, base-$b$ palindromes are equidistributed across residue classes modulo $d^2$ for square-free $d$ up to $x^{1/4-\varepsilon}$, with a strong logarithmic saving in the error term. This estimate underpins recent advances in understanding the asymptotic distribution of constrained palindromic numbers, most notably the infinitude and density of square-free palindromes (Johnston et al., 11 Jan 2026).

1. Formal Statement of the Equidistribution Estimate

Let $b \geq 2$ be a fixed base. Define $P^*_b(x) = \{ n \leq x : n$ is a base-$b$ palindrome and $(n, b^3-b) = 1 \}$, the restricted set excluding palindromes with trivial local obstructions. Let $$P_b^*(y;a,d^2) := \#\{ n \leq y : n \in P_b^*(x), n \equiv a \pmod{d^2} \}$$.

Proposition (Tuxanidy–Panario, Prop. 10.1): For any $\varepsilon > 0$ and any $A > 0$, there exists $x_0 = x_0(b, \varepsilon, A)$ such that for all $x \geq x_0$:

$$\sum_{\substack{d \leq x^{1/4-\varepsilon} \ (d, b^3-b) = 1}} \mu^2(d) \sup_{y \leq x} \max_{a \in \mathbb{Z}} \left| P^*_b(y;a,d^2) - \frac{1}{d^2} \# P_b^*(y) \right| \ll_{b,\varepsilon,A} \frac{ \# P_b^*(x)} {(\log x)^A}$$

That is, for all square-free $d$ coprime to $b^3-b$ with $d \leq x^{1/4-\varepsilon}$, the palindromes in $P_b^*(x)$ are distributed nearly uniformly among the $d^2$ residue classes modulo $d^2$, with a power-saving in the logarithm of $x$.

2. Structure of the Error Term

The error is given by:

$$\sum_{\substack{d \leq x^{1/4-\varepsilon} \ (d, b^3-b) = 1}} \mu^2(d) \sup_{y \leq x} \max_{a \in \mathbb{Z}} \left| \sum_{n \in P_b^*(y)} \left( \mathbf 1_{n \equiv a \pmod{d^2}} - \frac{1}{d^2} \right) \right| \ll_{b, \varepsilon, A} \frac{ \# P_b^*(x) }{ (\log x)^A }$$

The implied constant depends on $b$, $\varepsilon$, and $A$ but not on $x$. The use of the square-free indicator $\mu^2(d)$ ensures that only square-free $d$ appear. The result provides a uniform estimate in $y \leq x$ and over all residue classes $a$, with an error that is arbitrarily small relative to the main term by taking $A$ large.

3. Methods and Proof Outline

The proof comprises three main analytic steps:

• Fourier Expansion of the Palindrome Indicator: The defining property of a palindrome allows the set $P_b^*(y)$ to be encoded via a short exponential sum over digital variables. Classical Fourier analysis is then used to express the indicator of $n \equiv a \pmod{d^2}$ as an additive character sum, facilitating manipulations modulo $d^2$.
• Large Sieve for Square Moduli: The proof's crux is the application of the Baier–Zhao large sieve inequality tailored for square moduli, bounding sums of the type:

$$\sum_{d \leq D} \sum_{a \pmod{d^2}} \left|\sum_{n \in P_b^*(x)} e\left(\frac{an}{d^2}\right) \right|^2$$

for $D \leq x^{1/4-\varepsilon}$. A detailed analysis of the Fourier coefficients specific to palindromes ensures that the $L^2$-average over all $(a, d)$ is $\ll_\varepsilon x (\log x)^{O(1)}$.

• Transition from $L^2$ to $L^\infty$ Bound: Standard arguments (Cauchy–Schwarz or Gallagher’s lemma) convert the $L^2$-bound into the desired uniform (sup-norm) bound over all residue classes and $y \leq x$.

4. Foundational Definitions and Supporting Results

Base-$b$ palindrome: An integer $n$ is a palindrome in base $b$ if its base-$b$ digit expansion $n = \sum_i n_i b^i$ satisfies symmetry: $n_i = n_{N-1-i}$.

Restricted set $P_b^*(x)$: To prevent trivial local obstructions (e.g., failure of equidistribution for divisors of $b^3-b$), palindromes are restricted to those with $(n, b^3-b)=1$.

Relevant counting functions:

• $$P_b^*(y;a,d^2) = \#\{ n \leq y, n \in P_b^*(x), n \equiv a \pmod{d^2} \}$$
• $\# P_b^*(y) \approx \sqrt{y}$ for large $y$.

Tools:

• The large sieve for square moduli, in the form refined by Baier and Zhao, provides critical estimates for exponential sums over sparse sets like palindromes.
• Standard Fourier analysis, notably the expansion of indicators for congruence classes as sums over additive characters.

5. Application in Analytic Frameworks

In the context of proving the infinitude of square-free palindromes, the estimate is crucial in the Möbius inversion method, which expresses the count of square-free palindromes as:

$$Q_b^*(x) = \sum_{n \in P_b^*(x)} \mu^2(n) = \sum_{d \leq \sqrt{x},\ (d,b^3-b)=1} \mu(d)\ \#\{ n \in P_b^*(x) : d^2 \mid n \}$$

The sum is split at $x^{1/4-\varepsilon}$:

• For $d \leq x^{1/4-\varepsilon}$, the equidistribution estimate replaces the count of $n \equiv 0 \pmod{d^2}$ among palindromes with the expected main term $\#P_b^*(x)/d^2$, up to a negligible logarithmic error. This uniformly small error ensures sharp asymptotics for the Möbius-inverted sum.
• For $d > x^{1/4-\varepsilon}$, the error term exceeds the threshold of the Tuxanidy–Panario estimate, necessitating alternate techniques—principally the hybrid $p$-adic/Archimedean van der Corput process.

6. Scope and Limitations

The equidistribution estimate holds only for moduli $d \leq x^{1/4-\varepsilon}$. For larger $d$, the method does not suffice, and exponential sum techniques (van der Corput differencing and Poisson summation) are required. The error savings are polynomially small only in $\log x$, not in $x$ itself, which is sufficient for isolating the main term but not for more delicate secondary analysis.

The coprimality requirement $(d, b^3-b)=1$ is essential; otherwise, local obstructions disrupt equidistribution. Numerical results indicate that true equidistribution may persist far beyond the proven range (for $d^2 \approx b^N/\text{poly}(N)$), but current large sieve techniques do not capture this due to the sparsity of palindromes.

7. Significance and Further Directions

The Tuxanidy–Panario equidistribution result stands as the strongest analytic input for controlling the "small square-divisor" regime in Möbius inversion approaches to palindromic problems (Johnston et al., 11 Jan 2026). It demonstrates that base-$b$ palindromes, save for explicit local exceptions, are essentially uniformly distributed modulo small square moduli. Subsequent advances may hinge on extending large sieve techniques or finding alternative analytic frameworks capable of handling sparser or more structured sets, with implications for broader classes of digital and combinatorial number-theoretic problems.