Laczkovich 1984 kemperman s inequality

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backward_closure: Proves the finite-witness description of backward closure and the sublevel propagation needed to bound the function on a subgroup.

continued_fraction_inputs: States the exact classical convergent facts used in the finite-seed lemma and identifies the printed recurrence’s subscript correction.

definitions: Fixes nondecreasing monotonicity, the additive subgroup, and the finite-sequence conventions used in Laczkovich’s proof.

lemma_1: Proves the discrete endpoint bound by all three dyadic induction cases, including n=1 and the shortest final branch.

lemma_2: Proves finite backward coverage using continued fractions, with an enlarged auxiliary constant that treats both convergent denominators.

positive_increments: Proves density of the irrational additive subgroup and writes every positive difference as Nc plus (N+1)d with positive subgroup steps.

rational_counterexample: Checks the factorial-denominator construction against the stronger max inequality and proves its failure of monotonicity on the rationals.

stronger_max_inequality: Proves that the stronger inequality has only the zero nonnegative solution on the subgroup generated by one and square root two.

theorem_1: Derives the unrestricted real-line monotonicity theorem by an affine restriction to the subgroup generated by one and square root two.

theorem_2: Proves monotonicity from finite backward boundedness and two finite progressions, with a bound independent of their lengths.

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M. Laczkovich, On Kemperman's inequality $2f(x)\le f(x+h)+f(x+2h)$, Colloquium Mathematicum 49(1) (1984), 109–115. DOI 10.4064/cm-49-1-109-115.

Source and version

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Results and complete proof chain

Theorem 1 proves that every real-valued function on the entire real line satisfying

$$2f(x)\le f(x+h)+f(x+2h)\qquad(x\in\mathbb R,\ h>0)$$

is nondecreasing, without a measurability, continuity, or boundedness assumption. This is the ordinary mathematical conclusion asked in Problem 1125; strict increase is not claimed.

The main argument is Theorem 2 on the countable subgroup $\mathbb Z\alpha+\mathbb Z$, for an irrational $\alpha$ with bounded continued-fraction partial quotients. Applying it on an affine copy of $\mathbb Z\sqrt2+\mathbb Z$ compares any two real points and proves Theorem 1.

The two essential mechanisms are separately reconstructed:

• Lemma 1 bounds by $10K/n$ the endpoint drop of a finite sequence on $\{0,\ldots,n\}$ that satisfies the inequality and is bounded by $K$. Its three dyadic cases include the smallest lengths.
• Lemma 2 uses convergents to place a whole subgroup half-line in the backward closure of a finite seed. It relies on the exact classical continued-fraction inputs.

The finite-witness and propagation proof turns the finite seed into an upper bound for the function. The density and positive-increment proof then connects two subgroup points by two finite arithmetic progressions. Lemma 1 applied twice, with a bound fixed independently of their lengths, forces monotonicity.

The source's two further conclusions have full separate proofs:

• Lawrence's rational-domain construction satisfies even the stronger max inequality and is not monotonic.
• The nonnegative stronger-max consequence shows that on $\mathbb Z\sqrt2+\mathbb Z$ the only such nonnegative function is zero.

These give eight complete proof components, including the two expanded elementary interfaces. The definitions and classical continued-fraction statement page receive no separate full-proof credit. General continued-fraction theory remains an explicit external input. No formal proof build is part of this source compilation.

Source corrections and scope

The printed recurrence on p. 113 has coefficient $a_i$ where its indexing requires $a_{i+1}$. More substantially, its step can use either $q_j$ or $q_{j-1}$, but the following calculation uses $q_j$ in both cases. Lemma 2 now tracks the chosen denominator and proves the needed comparison with the enlarged auxiliary constant $(K+1)^3N^2$, replacing the source proof's $(K+1)^2N^2$. The theorem and lemma statements are unchanged. The local repair, its exact inequality, and its effect on the finite seed are visible on the lemma page; it is not presented as an author erratum.

Lemma 1 states $n\ge1$, handles $n=1$ directly, and ends its final branch immediately when $n=2^k+2$; only larger $n$ require the next backward step. The max-inequality proof expands the source's one-sided-limit comparison with positive steps in the dense subgroup.

Historical pointers

The paper credits Kemperman for the measurable-function case and Lawrence and Segal for earlier partial work. Its references are:

• J. H. B. Kemperman, On the regularity of generalized convex functions, Transactions of the American Mathematical Society 135 (1969), 69–93.
• J. H. B. Kemperman, Problem 60, Aequationes Mathematicae 4 (1970), 248–249.
• J. Lawrence, On Kemperman's inequality, listed as “to appear” in this source.
• S. L. Segal, On a functional inequality of Kemperman, Colloquium Mathematicum 35 (1976), 91–95.

Those separate original papers are not inspected or compiled here. The statement of P1282 on p. 110 asks whether bounded partial quotients can be omitted from Theorem 2; it is a question posed in 1984, not a claim of present openness. The footnote on p. 109 reports a generalization of Theorem 1 by Laczkovich in General Inequalities 3, edited by E. F. Beckenbach and W. Walter (Birkhäuser, 1983), pp. 281–293. That separate contribution has not been inspected, and no comparison or complete-proof credit is claimed for it.

Bears on. Problem 1125.