Samorodnitsky's Inequality

• Samorodnitsky's inequality is a set of noise-smoothing and correlation bounds for Boolean functions and lattice events defined on product spaces.
• It employs tensorization and extremal reduction techniques to derive optimal constants and connects with strong data processing inequalities for Rényi divergences.
• The inequality has practical applications in coding theory, hypercontractivity refinements, and almost sure local limit theorems for independent random variables.

Searching arXiv for relevant papers on Samorodnitsky's inequality and closely related formulations. Samorodnitsky's inequality is the name used for at least two distinct families of results. In analysis of Boolean and product-space functions, it denotes a noisy function inequality controlling the $L_q$ norm of a noise-smoothed nonnegative function by the expected logarithmic $L_q$ norms of its conditional expectations over random coordinate subsets: for $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$, $q>1$, and $0<\rho<1$, there exists $\lambda=\lambda(q,\rho)<1$ such that

$$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$

In probability theory, the same name also refers to a correlation inequality for indicators of lattice level sets of partial sums of independent random variables. The two usages are related by a common extremal and smoothing perspective, but they concern different operators, different state spaces, and different applications (Abawonse et al., 9 Aug 2025, Weber, 2023).

1. Scope and principal formulations

The noisy-function formulation is stated on the hypercube and, in its generalized form, on arbitrary finite product spaces. The level-set formulation concerns correlations of events of the form $\{S_n=K_n\}$ for sums of independent lattice-valued random variables. This suggests that the phrase “Samorodnitsky’s inequality” is context-dependent and should be interpreted from the surrounding subject matter rather than from the name alone (Abawonse et al., 9 Aug 2025, Weber, 2023).

Variant	Ambient setting	Characteristic form
Noisy function inequality	$\mathbb{F}_2^n$ or $(\Omega^n,\mu^{\otimes n})$	$L_q$0
Level-set correlation inequality	Lattice sums $L_q$1	Bounds on $L_q$2

A separate source of terminological confusion arises from sharp inequalities for sums of bounded independent random variables. One such result gives improved bounds on $L_q$3, compares primarily with Hoeffding, Bentkus, Talagrand, and Pinelis, and explicitly notes that Samorodnitsky’s inequality is not mentioned by name in that setting; this suggests that Hoeffding-type lower-tail bounds should not be conflated automatically with the noisy-function or level-set meanings of the term (Dance, 2012).

2. Binary statement and product-space generalization

In the binary form, the inequality applies to nonnegative functions $L_q$4. The operator $L_q$5 is the noise operator on the hypercube with correlation $L_q$6, $L_q$7 is the conditional expectation obtained by averaging out coordinates not in $L_q$8, and $L_q$9 is chosen by including each coordinate independently with probability $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$0. The crucial base case is the one-dimensional inequality

$f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$1

from which the full $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$2-dimensional result is obtained by tensorization (Abawonse et al., 9 Aug 2025).

The 2025 generalization extends the inequality from $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$3 to arbitrary product spaces $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$4, where $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$5 is a full-support probability distribution on finite $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$6. Writing

$f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$7

the generalized statement is that for every $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$8 and $f : \mathbb{F}_2^n \to \mathbb{R}_{\geq 0}$9, there exists

$q>1$0

such that for all nonnegative $q>1$1,

$q>1$2

The same one-dimensional inequality remains the base case in this generality (Abawonse et al., 9 Aug 2025).

For $q>1$3, the optimal parameter is explicit. In the uniform case, letting $q>1$4, one has

$q>1$5

Two special cases are singled out: $q>1$6 Samorodnitsky determined the optimal $q>1$7 for the binary inequality when $q>1$8 is an integer; the later generalization determines the optimal value for any real $q>1$9, any $0<\rho<1$0, and arbitrary $0<\rho<1$1 (Abawonse et al., 9 Aug 2025).

3. Extremizers, tensorization, and the SDPI interpretation

The generalized proof has two structural components. The first is tensorization: the full $0<\rho<1$2-dimensional inequality is reduced to the case $0<\rho<1$3. The key tool is a general Minkowski inequality for composing norms and expectations in product spaces, and the resulting induction template is described as unifying tensorization for hypercontractivity, reverse hypercontractivity, and Samorodnitsky’s inequalities (Abawonse et al., 9 Aug 2025).

The second component is extremal reduction. To determine the optimal $0<\rho<1$4, the maximizer is shown to be a two-point function, equivalently the indicator of a point $0<\rho<1$5 with minimal $0<\rho<1$6. In the paper’s terminology, the ratio

$0<\rho<1$7

for mean-zero random variables $0<\rho<1$8 with range $0<\rho<1$9 is maximized by a certain binary distribution. This directly yields the formula for $\lambda=\lambda(q,\rho)<1$0. The same extremal mechanism is summarized in function language by the statement that the worst-case maximizing functions are always indicators of the atom of minimal probability, often described as “dictators” (Abawonse et al., 9 Aug 2025).

The generalized inequality also has an information-theoretic reformulation. The base case is equivalent to a strong data processing inequality for Rényi divergences: $\lambda=\lambda(q,\rho)<1$1 The optimal constant in this SDPI is exactly the same $\lambda=\lambda(q,\rho)<1$2, and the tight cases are identified: equality is attained only when $\lambda=\lambda(q,\rho)<1$3 or when $\lambda=\lambda(q,\rho)<1$4 is a delta measure at an atom where $\lambda=\lambda(q,\rho)<1$5 is minimal. This places Samorodnitsky’s inequality in direct contact with channel contraction and Rényi-divergence decay under noise (Abawonse et al., 9 Aug 2025).

A further point of distinction from classical hypercontractivity is explicit in the generalized treatment. Although the inequality resembles hypercontractivity in shape, it quantifies smoothing in a way that hypercontractivity cannot match, and the optimal $\lambda=\lambda(q,\rho)<1$6 is strictly smaller than what would arise from Riesz–Thorin interpolation applied to hypercontractive norms (Abawonse et al., 9 Aug 2025).

4. Coding-theoretic consequences

One of the central applications of the noisy-function inequality is to error-correcting codes. The abstract states a binary consequence: any binary linear code with minimum distance $\lambda=\lambda(q,\rho)<1$7 that has vanishing decoding error probability on the $\lambda=\lambda(q,\rho)<1$8 also has vanishing decoding error on all memoryless symmetric channels with capacity above some $\lambda=\lambda(q,\rho)<1$9 (Abawonse et al., 9 Aug 2025).

The generalized version extends this style of conclusion to non-binary alphabets. For a linear code $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$0 with weight distribution $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$1, if $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$2 is uniform on $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$3 and $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$4 is the output of the $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$5-ary erasure channel with erasure probability $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$6, then

$$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$7

This gives a direct inequality between the weight enumerator and conditional entropy (Abawonse et al., 9 Aug 2025).

A robustness theorem follows. If a family of linear codes achieves vanishing error probability for erasure probability $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$8, then, under mild distance assumptions—typically distance $$\log \|T_\rho f\|_q \leq \mathbb{E}_{S\sim \lambda}\log \|\mathbb{E}(f\mid S)\|_q .$$9—it achieves vanishing block error on any symmetric channel with Bhattacharyya parameter

$\{S_n=K_n\}$0

The paper describes this as a universal transfer principle: goodness on the erasure channel is robust to the choice of memoryless symmetric channel. The $\{S_n=K_n\}$1 case yields new bounds on the probability of undetected error, and the same method is connected in the paper to list-decoding, Markov processes, sharp threshold theorems, and secrecy (wiretap) codes (Abawonse et al., 9 Aug 2025).

5. Correlation inequalities for lattice level sets

A second usage of the name concerns sums $\{S_n=K_n\}$2 of independent lattice-valued random variables. Here the basic object is the correlation of level-set events,

$\{S_n=K_n\}$3

where the $\{S_n=K_n\}$4 take values in a common lattice $\{S_n=K_n\}$5. The 2023 paper states that its main result extends Samorodnitsky’s inequalities from the i.i.d. case to general independent, non-identically distributed sequences (Weber, 2023).

The relevant regularity parameter is

$\{S_n=K_n\}$6

with cumulative smoothness

$\{S_n=K_n\}$7

Under the conditions $\{S_n=K_n\}$8, $\{S_n=K_n\}$9, $\mathbb{F}_2^n$0, and $\mathbb{F}_2^n$1, the paper proves a sharp correlation inequality in which the normalized correlation is controlled by a term involving the product $\mathbb{F}_2^n$2 and separation penalties of local-limit type such as $\mathbb{F}_2^n$3 (Weber, 2023).

A simplified corollary is available when the smoothness is uniformly bounded away from $\mathbb{F}_2^n$4: if

$\mathbb{F}_2^n$5

then the correlation decays exponentially in $\mathbb{F}_2^n$6, together with a $\mathbb{F}_2^n$7-type term. In the i.i.d. case this recovers the characteristic decay pattern associated with Samorodnitsky’s original result. The paper states that the extension is essentially as sharp as possible for general independent sequences and matches Samorodnitsky when specialized to the i.i.d. setting (Weber, 2023).

The proof method is Bernoulli part extraction. Each $\mathbb{F}_2^n$8 is decomposed as

$\mathbb{F}_2^n$9

where $(\Omega^n,\mu^{\otimes n})$0 carries the non-Bernoulli part, $(\Omega^n,\mu^{\otimes n})$1, and $(\Omega^n,\mu^{\otimes n})$2 is Bernoulli. Summing yields

$(\Omega^n,\mu^{\otimes n})$3

so that conditioning on the Bernoulli part gives effective control over the probabilities of lattice points. This decomposition is the basis both for the correlation inequality and for the paper’s almost sure local limit theorem (Weber, 2023).

6. Applications to almost sure local limit theory and neighboring inequalities

In the lattice-probability setting, the correlation inequality is used to control quadratic forms built from the indicators $(\Omega^n,\mu^{\otimes n})$4. The paper states that the normalized sums

$(\Omega^n,\mu^{\otimes n})$5

converge almost surely to zero under the smoothness and variance assumptions, thereby yielding a very general almost sure local limit theorem for independent, square integrable lattice-valued random variables (Weber, 2023).

The noisy-function and level-set versions of Samorodnitsky’s inequality interact differently with adjacent theories. On the function side, the generalized tensorization framework places the inequality alongside hypercontractivity and reverse hypercontractivity, but the paper emphasizes that Samorodnitsky’s inequality provides a more precise quantification of noise smoothing for the coding-theoretic questions under study (Abawonse et al., 9 Aug 2025). On the probability side, the level-set inequality is presented as a sharp correlation estimate for lattice events, with direct consequences for ASLLT rather than for norm contraction (Weber, 2023).

A distinct neighboring development is the sharp lower-tail bound for sums of independent bounded random variables in which

$(\Omega^n,\mu^{\otimes n})$6

in the Poisson limit regime. That result improves Hoeffding in a special case and is optimal when the sum converges to a Poisson random variable, but the associated paper compares mainly with Hoeffding, Bentkus, Talagrand, and Pinelis and notes that Samorodnitsky’s inequality is not discussed there by name (Dance, 2012). This suggests that the name “Samorodnitsky’s inequality” should be reserved for the noisy-function and level-set contexts in which the literature explicitly uses it.